2D Entry & Exit Angles

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Most texts show that a sail develops maximum lift (or drive) when set at an angle of attack of about 15 or 20 degrees to the incident wind. What they are much less clear about, however, is that an appropriate angle of attack closely depends upon the draft (and the position of maximum draft) set into the sail in the first place, and why such an apparently large angle of attack is appropriate. The following discussion concerns two-dimensional sail theory. (Yup, it all changes when three-dimensional theory is addressed.)

The diagram shows wind that is incident on a sail. [I've modified my idiosyncratic definition of the entry angle to conform to the rest of the world. Thanks to Larry Robinson and Henry Farley who, er, prodded me in this direction.] The curvature of the sail causes the luff to point into the wind. As the curvature of the sail is changed, its entry angle changes, and the sail needs to be set at a different angle of attack, so that its luff points "ideally" into the local apparent wind. Notice that, although exaggerated in the diagram, the entry angle is usually slightly less than the angle of attack. The sail curvature, of course, is mainly measured by its draft and the position of its maximum draft. For simplicity, I assume that the curvature of the sail can be approximated by the arc of a circle. If we assume a sail with a draft of 10% -- ie the depth of the draft is equal to 10% of the length of the chord -- and a position of 50% -- the maximum draft is halfway along the chord -- then the entry angle is about 22 degrees on this assumption. If this sail is sheeted or set at an angle of attack of 20 degrees, then the luff is pretty much pointing exactly into the incident wind. I suppose that's why an angle of attack of around 20 degrees is found to give maximum lift for a sail with 10% camber.

The spreadsheet model (about 26kb) calculates the entry angle given the sail draft and position of maximum draft.  One graph illustrates the change in entry angle as the draft itself changes, and another graph shows the entry angle as the position of maximum draft changes. In both cases, the change is fairly dramatic, and indicates that the angle of attack of the sail must be very carefully adjusted as draft and draft position are changed. Well, you knew that, of course, but did you know or have a feel for how much? The graph, below, illustrates the entry and exit angles when the max draft is located at 40% of chord. Entry angle varies from about 11 degrees for a flat sail, to around 25 degrees for a full sail.

Graph of entry and exit angles given sail draft

Entry angle also changes with position of maximum draft.  For example, as the position of maximum draft moves 10% from, say, 40% to 30%, the entry angle increases by about 3.5 degrees. If you make a sail with maximum draft at 30%, say, you'll need to remember to sheet in by a further 3 or 4 degrees. Similarly, for every 1% the sail is flattened or made fuller, the entry angle changes by about 2.2 degrees. So if you flatten the sail from a draft of 10% to, say, 8%, then the entry angle reduces by about 4.5 degrees; you should sheet out a further 4 or 5 degrees to maintain optimum drive.

Now for the science bit. In the spreadsheet model, the formula for calculating the entry angle, assuming that the sail curvature is like an arc of a circle, is an approximation given as ATAN(2*draft%/max draft pos%).  That is, given a draft of 8%, say, and a position of maximum draft of 44%, the tangent of the entry angle is 0.16/0.44, and so the entry angle itself is about 20 degrees.

The third graph of the spreadsheet calculates the inverse problem -- how the amount of draft needs to change to maintain a constant entry angle, as the position of maximum draft changes.  This typically happens as the wind picks up, when the sail begins to "bag forward".  The position of maximum draft then moves forward as well, and as it does so, the sail's entry angle increases.  In order to control this, you would normally flatten the sail as the wind picks up.  The graph illustrates the situation when you are trying to keep the entry angle at 20 degrees.  Suppose the sail has a basic 8% draft cut into it, with max draft at 44% of chord, and hence starts off with an entry angle of 20 degrees.  As the wind picks up, imagine the draft moves forward until, in the extreme, it sits at 24% of chord.  In this case, the sail draft needs to be flattened to about 4.4% in order to maintain 20 degrees of entry angle.  You'd flatten the sail by adjusting the outhaul.  Of course, the outhaul only affects the bottom 40% or 50% of the sail, so the rest of the sail has to be flattened by increasing mast bend.

2005-12-18


2011 Lester Gilbert