The cross section of the keel
The keel has two functions, to resist the transverse force generated by the sailing rig and to steer the boat. Normally the need to steer makes a small demand on the keel but there are a few manoeuvres in sailing that make a high demand, for example, when rounding a mark from a reach to turn into wind. Then the keel must produce a large transverse force to make the turn as well as producing a force to resist the transverse force of the sails. Furthermore we need a keel that will produce a high transverse force with a very small drag.
We now know that the best shape for the cross section of the keel is a symmetrical aerofoil and not a flat board. The attraction is two fold, we can have a very low drag and the aerodynamic keel can work at relatively large angles of leeway before it stalls. The best data we have was gathered by the NACA between the wars in a nationally funded programme of experiment. It was carried out at a time when the biplane fighter with its braced, thin wings was giving way to the monoplane fighter. This influenced the choice of sections chosen for testing but the choices that were made are very suitable for keels. The data is there for everyone to use. Before we start we must recognise that the data was gathered in wind tunnels and is not automatically transferable directly from aeroplane sections to the sections of a yacht keel. Strictly, data is only transferable between systems having the same value of Reynolds number. This means that we must have an idea of the value of Reynolds number for our keel.
Reynolds number, , is evaluated from where is the chord, is the speed of the undisturbed water and is the kinematic viscosity of water all in consistent units. For a keel of 2 feet chord moving at 10 knots the value of is about . The NACA data has been collected for , and so we can use data for the same Reynolds number as our keel.
NACA recognised that the performance of an aerofoil is affected by the roughness of its surface and, in particular, by the roughness of the surface near to the leading edge. They used models of 2 feet chord made to high standards of profile and finish and for each section made a special test with grains of carborundum 0.011² across spread thinly (5 to 10% of the area) over the first 8% of the chord ie over about 2². The major effect of this added roughness on NACA section 0012 is to reduce the angle of attack at which the section stalls from about 16° to about 12° and to reduce the lift coefficient from about 1.5 to about 1. This is a serious reduction in desirable characteristics for the want of a good finish. This means that the surface finish is important especially over the first 10% of the section and should be as good as can be maintained over the rest. One must also recognise that the profiles of the test sections were made very accurately. It is no accident that the specification of the wings of a high performance glider includes a statement of the accuracy of the profile expressed as the error in the profile over any six-inch square of the surface of the wing and that error is in thousandths of an inch. The aerodynamic keel should be made to a high standard of surface finish and have an accurate profile.
So what does NACA 0012-64 look like? I have drawn, in figure 69, four of the symmetrical sections on which NACA based whole families of asymmetrical sections. These symmetrical sections are candidates for use for both the keel and the rudder.
What we want from both these devices is that they produce a large lift for little drag and we want the stall to be gentle at a high angle of attack.
In order to choose we must look at the profiles. The critical region is the first 25% from the leading edge. The stall occurs when the flow on the upper face breaks away from the section. So what initiates the breakaway? The breakaway occurs when the curvature of the upper surface is too great for the water flowing over it to follow the profile, ie, when the pressure difference between the free stream and the profile is too small for the essential centripetal force to be generated.
There are constraints on aerofoil shapes. What we want is a shape that is made up of smooth curves. The way to get smooth curves is to use mathematical curves and the most simple is the parabola.
In figure 70 four parabolas give the basic shape for a symmetrical section. Aerofoils with pointed noses have cost several lives because they stall totally and suddenly. It is essential to blend the two left hand parabolas together to give a nose. If a circular arc is fitted to touch the left hand parabola and go through -4 the shape closely resembles the aerofoils given in figure 71. There is no end to the ingenuity that can be expended on these shapes especially for asymmetrical ones. But no matter how clever the designer may be the wing or keel has to be manufactured and extreme refinement is pointless. The important point is that some shape such as this is unavoidable if it is to operate as a wing.
In figure 72 I have drawn the nose of NACA 0012 64 making angles of attack of 0°, 2° and 5° together with representations of the likely flow patterns. In each case there is a flow line that “hits” the nose (It is called the stagnation line.) and it divides the flow into streams that go over and under the aerofoil. On each profile I have shown the 10% point with an arrowhead. It is desirable for the flow to follow the surface everywhere. On the underside of the aerofoil this is inevitable because the solid surface can offer passive resistance to any pressure exerted on it. This cannot be the case over the upper surface where the flow must be made to bend downwards by a pressure gradient from the free stream to the surface. If this is insufficient the flow will separate from the aerofoil. The critical point will occur where the radius of curvature is greatest and the diagrams show that, as the angle of attack increases, a region just behind the nose is the point where the greatest radius of curvature develops. At some angle of attack separation will occur and the aerofoil will stall. These simple diagrams show why NACA were only interested in roughening the first 8%. If the surface is rough the air near to the surface will slow down and the pressure will rise reducing the inward pressure difference.
However it is the rudder that is most likely to let us down by stalling because it is too often used to balance the boat instead of balancing it by adjusting the rig and, on small boats at least, the rudder can be moved too quickly. Its real function is to control the angle of attack of the keel to resist the transverse force from the rig and, additionally, by increasing or decreasing this angle of attack of the keel, to steer the boat. In a well-balanced boat neither the keel nor the rudder works at a high angle of attack in normal beating, reaching or running but, at a change in direction, eg at a mark, it is quite possible to throw the rudder over to produce an excessive angle of attack and make the rudder stall. It is much better to feed the rudder on so that the boat can start to turn at a progressively greater rate without stalling the rudder and keep up the boat speed.
The keel does the same job on a
yacht as a wing does on an aeroplane. Aeroplane wings have all sorts of
plan-forms. Far and away the most common is the straight taper. This was used
for the American Mustang fighter aeroplane that had a symmetrical
section like that which must be used for yachts. The Mustang was capable
of escorting bombers to
However there is always the question of aspect ratio to consider. The aspect ratio is the length of the keel divided by its average width. For aeroplanes it is known that the efficiency of the wing, as measured by lift over drag, increases as the aspect ratio increases. There is no reason to suppose that the keel behaves in any different way. There are constraints on keels because high-aspect ratio keels are long and require a large depth. As a result the keels on boats are often of low aspect ratio, about 1.5 to 1 where the two wings of an aeroplane would have a ratio of at least 6 to 1 to give a ratio for one wing of 3.
It is worth looking at the proportions and aspect ratio of a keel. The graph in figure 73 is for a keel of 20 square feet. The range of mean chord is from 2 foot to 10 feet and this covers most keel shapes. The scales are equal so the proportions of any keel shape is easily seen eg for a length of 5 and a mean chord of 4. Clearly a keel of 10 feet length and 2 feet mean chord looks very slender to carry a bulb. On the other extreme a length of only 2 feet and a mean chord of 10 feet starts to look very unsuitable for an wing. The useful shapes lie between these two. I have drawn representative profiles for different aspect ratios in figure 74. When looking at these it must be remembered that as the length of the keel increases for a given boat the weight and size of the bulb may be reduced. Bulbs for shallow keels can be very heavy and very large and not at all easy to fit.
I have pointed out that an aeroplane wing generates two vortices centred on the wing tips. The vortex that might have been generated at the top of the keel is suppressed by the hull. But the keel will generate one off its lower tip and the lower the aspect ratio the greater will be the drag created by this vortex. In aeroplane design, junctions, say between a fuselage and a wing, are often faired or filleted to enclose a volume that would otherwise contain a vigorous eddy or many such eddies. We have to attach a bulb to the outer end of the keel and, if this is properly designed as a round streamlined shape, it can fill the centre of the vortex and reduce the drag just as a fairing does.
It is quite easy to make a calculation for the area of a keel from aeronautical data. We can use the expression for lift:-
It can be used twice, once to calculate a magnitude for the maximum transverse force produced by the sailing rig and again for the force generated by the keel. All these symbols must be given practical values.
In doing so one must recognise that the of drag of an accurately-made, smooth, symmetrical fin when it is working under a sailing boat will be very low indeed compared with the friction drag of the hull and the drag due to wave making and, as a result, being generous with the area of the fin incurs no discernable penalty. This means that we can make sensible decisions about these values and assess the outcome in the light of these decisions and any other simplifications.
Let me start with the transverse force to be resisted by the keel. The transverse force on the whole boat is caused by the action of the rig in producing lift and its consequent drag plus the drag on the standing rigging and the hull and its appendages. If we find the maximum conceivable force produced by the rig it will cover the other drags as well.
We have seen that the greatest coefficient of lift for a single section of a sail is about 1.3. Given that real sails will be triangular and twisted it might be better to use half that figure say 0.7. For any boat the total sail area and the highest wind speed for which the boat can carry its rig will be known. So a value for the maximum transverse force can be calculated.
The keel generates a transverse force to resist this transverse force by moving at some angle of attack to the flow of water at boat speed. If the boat sails with a full rig in the highest wind it can sustain, the speed of the boat will probably be some fraction, eg 0.8, of the “hull” speed ie 1.44Öl knots where l is the waterline length in feet.. Then of course a coefficient of lift for the fin must be selected. The graph of lift coefficient against angle of attack is well known because it is always a straight line passing through 0.8 at 8°. One must decide on an acceptable angle of leeway. One cannot work up to the stall at say 10° and must have some margin in hand for steering. An angle of about 4° would be reasonable based on the fin resisting the transverse force on its own without help from the hull and some might want to reduce the leeway still more. If we stick with 4° the coefficient of lift will be 0.4.
I like to plot graphs to see what all this means not make isolated calculations. I could draw a family of graphs of the ratio of the area of the keel to the area of the rig for a range of lengths of boats at a different wind speeds.
For air density is 0.0765 lb/cu foot. Density of water = 62.2 lb/cu foot. 1 Knot = 1.69 feet per second. Then:-
For the rig maximum transverse force =
where W is the wind speed in knots and is the rig area in square feet.
For the keel lift =
Where l is the waterline length of the boat in feet and is the area of the keel in square feet.
These two quantities are equal and it follows that :- =
This is the graph of the area ratio against waterline length for various maximum speeds for an angle of leeway of 4°. There will be a similar graph for every angle of leeway.
If this ratio is evaluated for a given boat it may well be that the resulting keel is not big enough to contain the structure needed to carry the bulb. The area can then be increased resulting in a reduced leeway. It is very fortunate that the drag of these smooth fins is very low. It makes the design so much easier.