The theory of downwash starts by noting that you only get downwash when you have lift. No lift, no downwash. In fact, the Circulation theory of lift says that, if you want to calculate the amount of lift, you calculate the downwash. Same thing. The Momentum theory of lift says something very similar. The Lifting line theory of lift modifies the simple circulation theory with a little extra detail on the downwash due to the wing tip vortices.
The sketch, based on Garrett (fig 3.6) shows a 10% cambered plate inclined at a substantial angle of attack to an oncoming flow. The flow is a "potential" flow -- that is, it is theoretically what happens around a wing (sail, fin) section, but you'll not see it in practice because it ignores the boundary layer and flow viscosity. Nevertheless, it is indicative -- it shows how the foil displaces the flow downwards to give downwash at the trailing edge, while also showing the significant upwash that is created at the leading edge.
This diagram of potential flow around a cambered plate is a little misleading, because it isn't showing the almost certain fact that the plate is stalled -- there will be turbulence behind the plate and not the nice streamlines being shown. A 10% cambered plate achieves maximum lift at around 15 degrees angle of attack, and by 26 degrees of attack will certainly have entered the stall region. And, a downwash of 26 degrees isn't something you'd normally see from a jib or mainsail.
Here is another "potential" flow sketch, from Bittle (picture 9-6b). This is a little less "potential", since Ivor produced it using a Hele-Shaw flow visualiser. This device uses water, and hence there is viscosity involved, so the flow lines are a little more realistic. The proportions and angles simulate an IOM close-hauled. What is interesting here is how the 2-D flow shows very little downwash overall -- the flow lines top and bottom are pretty much horizontal, and in particular they run horizontally almost immediately behind the mainsail leech, not "extending" downwards at all. The jib does show modest downwash over the lee of the mainsail.
NACA TR 648 looks at downwash behind a wing, but interestingly does this for different wing tapers. This is useful, because our triangular sails have high tapers, while "normal" wings, fins, or rudders have lower tapers. In theory, a perfectly pointed sail has an infinite taper, but something around 1:7 works fine. If an IOM fin is, say, 120 mm at the root and 80 mm at the tip, it has a taper of 1:1.5. An elliptical planform can be considered to have a taper of around 1:1.7.
In order to use the data from TR 648, the spreadsheet converts the taper, 1:t, to a decimal taper ratio. So a taper of 1:2 becomes a ratio of 0.5. The spreadsheet then can produce a trend line of best fit, relating the taper ratio to a measure of how much the downwash changes due to taper. The TR 648 paper gives a range of constants, "K", and the spreadsheet calculates "delta e", the change in downwash due to taper:
Then, TR 648 looks at downwash depending upon the aspect ratio of the wing. In theory, explained on the Momentum theory of lift page, downwash depends on two main factors: lift coefficient, and aspect ratio. In theory, the constant "K" for a "perfect" rectangular planform wing is pi/2 radians, around 36.5 degrees.
An IOM mainsail has AR approximately 6.8 and the jib AR is approximately 6.6. Usually, aspect ratio is the ratio of the span of the device to its average chord. An IOM No.1 jib has a span of about 1250 mm, and a foot of 380 mm, giving an "average" chord of 190 mm. An IOM fin, 360 mm long and with an average chord of 100 mm, has AR = 3.6.
Some texts point out that a sail or fin is but half a wing, and that to obtain the "true" aspect ratio, the AR of the sail or fin should be doubled. This is often "explained" by the idea that the water surface acts as a "mirror" for the sail or fin. It might be a little easier to understand if we instead looked at the gap between the foot of the sail and the hull, or the end-plate that the hull provides for the fin root. In each case, if the foot of the sail is well sealed against the deck, or the hull forms a good end-plate at the fin root, then the vortex which normally forms at that end of the lifting surface is so diminished that the sail or fin shows a decrease in drag, and a decrease in downwash, "just as though" its aspect ratio had been increased. That's it. So if your sail is perfectly sealed on deck, or the hull in the vicinity of the fin root is quite flat and wide, by all means double the AR of the lifting surface. Otherwise you'll need to consult one of the expert texts to see how much you can inflate the AR to take account of a "small" gap to deck, or a "relatively" flat hull bottom.
Given the taper and the aspect ratio, the spreadsheet estimates a "delta e", the change in downwash for unit lift coefficient. Typical values for a fin would be around 13 degrees downwash per unit lift coefficient, and perhaps 10 degrees for a mainsail.
The example spreadsheet data here shows a sail with a taper of 1:7 and an aspect ratio of 6.8 -- an IOM mainsail, perhaps. The taper ratio is thus about 0.14, and the predicted downwash per unit lift coefficient is shown as 10.03 on the left-hand side, or rounded to 10 in the body of the table, where other downwash angles are estimated for a range of lift coefficients.
The downwash behind the wing -- sail, fin -- depends on the amount of lift the device is generating. More lift means more downwash. If we know the coefficient of lift, we can estimate the downwash angle.
For a fin operating with Cl = 0.2, say, the downwash at its trailing edge might be around 2.7 degrees, quite modest. For a mainsail operating with Cl = 1.2, downwash at the leech might be around 12 degrees. More interestingly, the jib would give a similar downwash, and this helps explain why the mainsail is sheeted about 10 degrees more tightly than the jib when beating, because it is operating in the jib's downwash field.
Note that these downwash estimates are for the "middle" of the fin or sail, and do not apply near the root or foot, or near the tip or head. There are vortices in these extremes that make it impossible to talk about downwash as though it was some kind of constant in those regions.
©2011 Lester Gilbert