by Lester Gilbert
A while back, I realized I didn’t really know whether I was
maximizing drive or maximizing efficiency when beating to windward. We’re nearly
at the end of quite a journey, having looked at the lift-to-drag ratio as a
measure of efficiency, at the course theorem, which explains how our ability to
point depends on efficiency, and at VMG and the heading that maximizes our
progress towards the windward mark. Finally, we are now ready to pull it all
together to find out what we need to do to set the rig for the optimum pointing
angle for maximum VMG.
Ross Garrett’s “The
Symmetry of Sailing” (1987, Fig. 3.35) tells us that optimal velocity made
good (VMG) when beating to windward occurs in a curious and counter-intuitive
position on the lift-to-drag curve—when drive is higher than that seen at best
efficiency, but not quite as much as when drive is maximized.
Let’s have a look at the lift-to-drag curve Ross uses.
Previously, we saw a plot of the lift-to-drag ratio for various boat headings.
We now need a version of this, but instead of the ratio, we are going to plot Cl
(coefficient of lift) versus Cd (coefficient of drag), and make a note on the
resulting curve of the heading that gave us that particular pair of Cl and Cd
points. Figure 1 is an example of this kind of lift-to-drag curve.
On this curve, we have shown two interesting points. One
point is where Cl is a maximum, i.e., where this curve peaks; maximum Cl is
around 1.6 when the heading is around 39 degrees. Another point on the curve is
the place where the Cl/Cd ratio is a maximum. What is important here is how we
got to see this—by drawing a tangent to the curve from the point of zero lift
and zero drag. The tangent touches the curve at a heading of around 15 degrees,
where the Cl is about 1.0 and the Cd about 0.03, giving a lift-to-drag ratio of
Previously, we noticed that sailing the boat at a heading of
15 degrees is not going to win too many races. We might be maximally efficient,
but our VMG would be close to zero. Let’s take a look, though, at what this
heading of 15 degrees looks like if we plot it from the lift-to-drag curve. The
result is shown in Figure 2. What is important is how we visualize this—by
drawing a perpendicular to the tangent line that corresponds to the point of
maximum Cl/Cd. This perpendicular conceptually shows us where to point, relative
to the true wind direction and the windward mark, if we wish to sail at maximum
efficiency. And we can see that it would be pretty hopeless pointing so high.
Now, let’s draw the heading that was suggested to us by the
VMG polar and see where that tangent touches the lift-to-drag curve. The
resulting diagram is shown as Figure 3.
It turns out that the point on the Cl-to-Cd curve that
corresponds to the optimum VMG heading is somewhere above the maximum efficiency
point, and somewhere below the maximum lift point. The lift-to-drag we are
seeking is nowhere near the point of maximum lift-to-drag efficiency and is
about equally far away from the point of maximum lift. Figure 4 illustrates
that, in some sense, it is “halfway” between the two.
My takeaway from this is that we are trying to set up the
boat so that it is in the VMG groove when beating to windward. We know that a
well-behaved boat sails in its groove “by itself”; that is, it sails happily at
some heading where it is balanced. The trick is to make its comfortable balanced
groove a VMG groove, and not a continually pinching groove or a needlessly
footing groove. Our analysis tells us that a VMG groove does not require us to
set the rig at maximum efficiency, and it also does not require us to set the
rig for maximum drive; but it does require us to set the rig pretty much
“halfway” between the two. Good luck with that!
A note on efficiencyFrom this discussion, some might think that efficiency, or maximum efficiency, is therefore not particularly important. Not so! What we are saying is that you do not need to set up your rig to work at its maximum efficiency in order to achieve your boat’s optimum VMG, but you certainly must have an efficient rig and boat to be able to point at all. According to the course theorem, the more efficient your boat and rig, the better you will be able to point while keeping up boat speed. In this case, your optimum VMG will change because of your improved efficiency to give you a higher pointing optimum heading.
AcknowledgementsGraham Bantock made some helpful comments on an earlier draft of the article.
©2023 Lester Gilbert