Wingmast Aerodynamics
Before I get into the aerodynamics, a cautionary note about the limitations of the methods I've used. The theoretical methods I have are strictly two dimensional. That is, they apply to the cross section of a shape that is infinite in length and rigid. A real soft sail is inherently a three dimensional, flexible problem, since we all know that the shape of the sail is affected by the tensions up and down the sail, as well as the tension in the streamwise direction. So you really have to combine the material strains and the aerodynamics of the whole rig to get the true picture. But 2D isn't a bad approximation and it has a lot to say about the cross section shape.
The other limitation is that the programs I have cannot really handle separated flow. But they can identify the onset of separation. For separation on the lee side, I've assumed that the lift will be reduced in proportion to the amount of surface that's separated. This is extremely crude, and its only real value is indicating whether the surface's stall characteristics will be gradual or sharp. For separation on the windward side, I've assumed that the pressures will be constant from the onset of separation until the same pressure is encountered farther back on the surface. This is another gross approximation, and I've no independent data to justify it. At best, both these methods of modeling the separation are qualitative in nature, in that they show the trends that can be expected. But they are not to be taken as accurate predictions. So take things with a grain of salt, and I hope this makes these exotic rigs a bit more understandable.
The basic features of the flow around a wingmast-sail combination are sketched in Figure 1. As the wind approaches the leading edge, part of it will pass to leeward, part to windward, and the dividing line between the two will come to a complete stop near the front of the mast. This is what we call the stagnation point, and it's where you'll find the highest pressure on the whole airfoil Everything else is downhill from here. As the air whips around the leading edge, it speeds up tremendously because of the low pressures needed to make it bend around the sharp curve. So in a short distance, it goes from dead stop to the highest velocities it'll see on the whole airfoil. But it has to slow down to get back to something near ambient pressure by the time it gets back to the trailing edge.
A computed velocity distribution for a typical wingmast-sail combination is shown in Figure 2. The velocities are generally faster on the leeward side, and slower on the windward side. A guy named Bernoulli proved that when air isn't sustaining any losses, which is pretty much true of the air everywhere but near the surface or in the wake, high velocities mean low pressure, and vice versa. So the pressure distribution has much the same shape, with low pressure on the lee side and higher pressure on the windward side. The area between these two curves represents an unbalanced force acting at right angles to the flow - the "lift" force. We want the two curves to be as far apart as possible to give us the most lift. But they have to come together at the end.
Now it's a strange but true fact that the air that is immediately in contact with the surface sticks to it and does not move! This air drags on the volume of air going by just outside of it, which drags a little less on the volume outside of that, and so forth. The result is a thin boundary layer, in which the airflow goes from zero velocity to whatever the free stream velocity is just outside the boundary layer - like moving from shore into a swift river. If you add the thickness of the boundary layer to the surface, you effectively get a new shape that the air has to flow around, and this determines the velocities of the flow throughout the rest of the flowfield. These velocities, through their corresponding pressures, have a profound influence on the boundary layer. So there's this intimate dance between the two.
It's another strange but true fact that all of the forces on a body are determined by the boundary layer, because without it, the pressures would add up in such a way that there'd be no net force. To get good performance, you have to stress the boundary layer hard. But push it too far, and it'll let go and your rig stalls. So the art of airfoil design turns out to consist of manipulating the velocity distribution in order to manage what's going on in the boundary layer.
At first, near the stagnation point, this boundary layer is very thin. But as you trace the flow downstream, the boundary layer gets thicker because the air slowed by the upstream surface continues to drag on the air farther out, and the air nearer to the surface is slowed down some more. So the effect of the surface diffuses outward into the flow as it sweeps along the airfoil. At some point, the boundary layer can't maintain this smooth state of affairs because it becomes unstable, and eddies start to appear. This is known as the transition from laminar (the smooth flow) to turbulent flow. A turbulent boundary layer is much thicker than a laminar flow, because the eddies are taking big chunks of low velocity air from near the surface, and throwing them some distance away from the surface. They are also bringing some of the higher velocity air from outside down closer to the surface. This higher velocity air gets slowed down, naturally, so this causes the "skin friction" of the turbulent boundary layer to be higher than the laminar boundary layer. But the turbulent boundary layer is not all bad, as you'll see later.
There are something like four ways the flow can transition naturally to turbulent flow. Two (cross flow and attachment line instabilities) only apply to swept wings at high speeds. Another, in which small disturbances moving downstream in the flow get amplified until they turn unstable and kink up into eddies, can occur in high winds. Delaying this kind of transition (Tollmein-Schlichting instabilities) is what the famous NACA laminar flow airfoils (the 6-series designations) were designed to do. But, given the size of our rigs and the speeds at which we operate (especially in light winds), the laminar boundary layer is stable enough that we will almost certainly see the transition occurring after laminar separation.
If the pressure is decreasing (which means the air velocity outside the boundary layer is increasing), then it sucks the disturbed air along, and things keep flowing smoothly. If the flow is running into increased resistance, as when the pressure is increasing, then air next to the surface can finally run out of steam and get pushed backward. When this happens, the flow separates from the surface. Now a laminar boundary layer is much more fragile in this respect than a turbulent boundary layer. So we want to maintain a fair amount of laminar flow so as to keep skin friction low, but we want the flow to be turbulent as the air slows down heading back to the trailing edge. By using the turbulent boundary layer's ability to slow down more, we can use higher velocities up front and get more lift. A laminar boundary layer is a little like driving on ice. You don't dare go too fast because you can't slow down quickly. A turbulent boundary layer is like driving on wet pavement - you've got better braking, so you can go faster without breaking loose. We want to get off the ice and onto the wet pavement before we have to start braking hard or we'll lose it!
Figure 3 shows what we want to happen when the laminar boundary layer separates. Right after it separates, the pressure becomes constant, which is characteristic of all separated flow, and the flow becomes unstable. Soon eddies form and the flow becomes turbulent. When this happens, the pressure increases at pretty much the maximum rate that a turbulent boundary layer can sustain. If this pressure increase/velocity reduction intercepts the pressure/velocity dictated by the shape of the surface itself, then the flow reattaches and forms a laminar separation bubble. If the two pressure curves don't intersect, the flow stays separated and the airfoil is stalled. Within the bubble, the air is recirculating - flowing backward next to the surface from the attachment point to the separation point. In marginal conditions, the bubble might cover a large portion of the surface. But this causes a lot of drag and is very fragile. A small increase in lift and poof! stall. We want a short, robust separation bubble that is positioned where we want it.
So, for a wingmast airfoil, you want to have high velocities on the lee side for high lift, but you don't want a really sharp pressure spike at the leading edge. This would mean a steep adverse pressure gradient there, and stall due to laminar separation without reattachment (the curves won't intersect). Instead, when stall occurs, you want it to start because you've stressed the turbulent boundary layer too much at the trailing edge, and you want the turbulent separation point to move forward gradually as you increase the angle of attack. This makes for a gentle stall and a forgiving sail rig.
Now lets take a look at Figure 2 again, but this time focusing on the windward surface velocities. Notice that there's a peak velocity near the leading edge, and a steep drop to the joint between the mast and sail. This adverse pressure gradient will lead to laminar separation, hopefully followed by turbulent reattachment, and finally turbulent separation before the flow gets to the joint. But look what happens after the joint. The velocity is increasing all the way to the trailing edge, which means the air is being sucked along. It isn't fighting an uphill battle the way it is on the lee side. So once across the joint, the flow reattaches again. This forms a turbulent separation bubble near the mast-sail intersection. This isn't good, but it isn't disastrous, either. It's a price we have to pay for the symmetry of the wingmast.
If the pressure increase on the mast's windward surface is too great, which happens at low angles of attack, the flow separates and doesn't reattach. This pretty much sets the minimum angle of attack for that shape. Small wingmasts have a much shorter distance between the peak near the leading edge and the joint at the sail. So the pressure increase is much steeper for small wingmasts. This means that a small wingmast has a narrower range of useable angle of attack between separation on the windward surface a low angles, and stall at high angles. Of course, the mast can be rotated to help alleviate this. But the fact remains that a small wingmast will have a narrower "groove" than a large wingmast. This will make it more difficult to trim well, and it will be more affected by changes in the local flow angles along the mast, such as from gusts or wind shear. But it may also be lighter in weight and have less drag when it's in the groove.
OK, so how do you come up with a shape for a wingmast? The best way is to design a mast and sail shape together, starting with the kind of pressure distribution you need. You want it to have characteristics which cause the laminar separation point to move smoothly from well aft on the airfoil at low angles of attack, to near the leading edge at high angles of attack. This will help to avoid leading edge stall due to laminar separation, and make for a progressive stall due to turbulent separation. Unfortunately, this requires a computer program to calculate the resulting shape.
Another approach I've come up with is based on modifying an existing airfoil. The conventional way of looking at airfoil aerodynamics is to represent the airfoil as a mean camber line plus a symmetrical pressure distribution. This was a good way of calculating the velocities in the days before computers, because you could calculate each one separately and superimpose the results. But another way of looking at it is to consider each surface separately. The velocity at a given point is heavily influenced by the airfoil's curvature at that point. The more convex the surface, the more negative the pressures will be in order to bend the flow. Likewise, a concave surface will tend to have high pressures or an adverse pressure gradient in order to bend the flow the other way. So if we base the wingmast-sail shape on the lee side contours of an existing airfoil, the characteristics should mimic that airfoil's characteristics to some extent. This approach works surprisingly well.
Figure 4 shows the steps in the process. First, select an airfoil that has the characteristics you want, especially near the leading edge. It should be fairly thick or cambered, because this will determine the draft in the sail shape. It should also have the characteristic that the lee side laminar separation point (transition) moves smoothly toward the leading edge as angle of attack is increased. Next, set the percentage of the chord you want to use for the mast, and mark that on the upper surface.
Now draw a line from the mast-sail joint to just below the leading edge. You'll want to place the end of the line so that it is perpendicular to the airfoil contour. If it's too far up, you'll get a sharp crease at the leading edge, and if it's too far down, you'll get an indentation. Finally, measure off the distances perpendicular from the line to the airfoil contour, and lay out points equally distant to the other side of the line. This forms a reflection of the part of the airfoil and completes the wingmast airfoil. That's all there is to it.
Let's work through a typical example. I picked the classic Clark Y airfoil, because it has been a proven performer over a wide range of conditions. I used the procedure above to create a family of wingmast/sail combinations, with mast sizes ranging from 10% of the total chord to 50% of the chord. The resulting airfoils are shown in Figure 5. With a larger chord, the wingmasts get physically thicker, and the mast rotation flattens out.
Velocity distributions for 10 different angles of attack are shown in Figure 6 for the case of the largest mast (50% chord). Note that the lee side velocities peak near the leading edge, and nearly the whole lee side has an adverse pressure gradient. This is typical of airfoils designed for low speeds, in order to give the flow the maximum distance to coast down from the maximum speed and to avoid any steep gradients that might cause the laminar separation bubble to "burst" (fail to reattach). Figure 7 shows the corresponding boundary layer displacement thickness to give a feel for just how thin it is. These thicknesses assume that all the flow is attached. The boundary layer computation can't handle separated flow, so it fails on the windward side approaching the mast/sail junction.
Figures 8 and 9 show the effect of changing mast size while keeping the angle of attack constant. The velocities over the lee side are nearly unchanged. But the big change is in the windward side. The windward suction peak is significantly smaller in magnitude for the larger wingmasts, and the steepness of the adverse pressure gradient is dramatically less.
I've marked the predicted separation points on these curves as well. At four degrees angle of attack, the upper surface is fully attached. At eight degrees, separation is just starting to set in on the upper surface, and as the angle of attack is increased, it will move forward on the sail and the stall will deepen.
The windward side is very interesting. The flow separates on the back of the mast because the flow is slowing down sharply as it approaches the mast/sail junction. As with the laminar separation bubble, the pressure is probably a constant in the separated region. So I've assumed that the flow reattaches when the velocity comes back up to the same level it was when it separated. This has the effect of chopping off the dip in the velocities there, and results in some loss of lift, as shown in Figure 9.
This loss is negligible at higher angles of attack. But at low angles of attack the rig suffers something akin to leading edge stall. Figure 8 shows what's happening. The velocities form a sharp peak as the air has to turn through a greater angle going around the leading edge at low angles of attack, particularly for the smaller wingmasts. The steep deceleration after this sharp peak causes the flow to separate early, and for the 10% wingmast, the flow doesn't reattach until almost the leech. And at lower angles of attack, it won't reattach at all. A soft sail would be luffing under these conditions, but a wingmast might not.
The next figures show the trends in these key boundary layer events as the angle of attack changes. They also show how these trends vary with the different mast designs.
The transition location (Figure 10) shows where the laminar separation is taking place. Transition and reattachment are assumed to happen closely after that. As the angle of attack increases, the adverse gradient on the lee side becomes progressively steeper (Figure 6). So the velocity reaches the slope needed for laminar separation at an earlier point on the airfoil. The opposite is happening on the windward surface. However, the movement here is not very great - the slopes are dominated by the proximity of the mast/sail junction and the leading edge, rather than by the angle of attack.
The same thing is happening with the turbulent separation points, just farther back (Figure 11). On the lee side, the flow is essentially fully attached through six degrees angle of attack, and doesn't really move too far forward until after 10 degrees. The windward side separation on the wingmast stays parked just upstream of the mast/sail junction. Things get a lot more interesting on the windward side when the reattachment points are shown as well:
Figure 12 has the same windward separation points as Figure 11. The distance between the separation lines and the reattachment lines shows the extent of the turbulent separation. The sudden growth in the separated region at low angles of attack for the small wingmasts is clearly evident. The large wingmasts are affected, too, but not nearly as much. Besides having a smaller separation region, the large masts also have lower velocities to start with, so the separation penalty for them is not so great.
This is shown in Figure 13. The top line, labeled "MCARFA" is the lift curve as computed, assuming that the flow is fully attached everywhere. At large angles of attack, I've simply reduced the lift in proportion to the amount of separation predicted for the lee side. At low angles of attack, I've modified the results by assuming that the pressure is held constant across the windward separation region.
The narrow groove of the small wingmasts is apparent here. The 10% chord wingmast performs well between six and eight degrees angle of attack. Above 6 degrees it starts to stall, and below four degrees it starts to suffer separation on the windward side. The large masts (175 and up) have the same stall characteristics, but don't suffer from a loss of lift at low angles of attack. 20% chord looks to be just a little on the small side, as it suffers from a modest loss at low angles of attack.
The drag penalty of the windward separation region is shown in Figure 14. I haven't added any drag increment for leeward separation. The curve marked "50% CdP" was obtained by integrating the pressures around the largest wingmast section. Getting the drag this way is a notoriously unreliable way to do it. The curve marked "50% Cd SY" uses the Squire-Young formula which is based on the characteristics of the boundary layer at the trailing edge. This is much like measuring the losses in the wake in the wind tunnel to get the drag, and is a much more reliable method. I typically plot both curves as sort of a quality check on the results. When the two are close together, I tend to believe the results more than when the differ.
The curves marked "50% Windward Sep" and on, are the Squire-Young drag results with an increment derived from integrating the pressures with and without the windward separation region. I was surprised to see that this gave a drag increment that was consistent across the whole angle of attack range, and differs mainly with the size of the wingmast.
The 10% chord wingmast starts to approach the results for the largest masts, so it appears that both very large and very small masts can be equally efficient. Provided that the smaller mast is kept in its groove. This may require constant adjustment of the mast rotation to get it to perform.
Figure 15 shows the result of putting the lift and drag effects together. The "50% (Raw)" curve assumes the flow is fully attached, while the "50% Sep" curve has both the lee side and windward side separation effects added. Again, the main difference between the different mast designs is their drag.
Finally, the sectional lift/drag ratio for the various masts is shown in Figure 16. Once again, the performance of the largest and smallest masts is similar at their peak. But the large mast maintains its performance over a very wide range of angles of attack compared to the small mast.
So that's the
story on wingmast-sail aerodynamics. It's not simple, but I hope that having
seen the real numbers, you've got a
better appreciation
of what makes these things tick. It's all theoretical, so if anybody out
there has some experimental data, I'd love to compare them with predictions.
See you on the race course.