# Asymmetrical Speed Polars

Discussion in 'Hydrodynamics and Aerodynamics' started by tspeer, Oct 10, 2012.

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### tspeerSenior Member

This is a branch of the discussion that started on the Jib question thread. In our last episode,

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### tspeerSenior Member

OK, I think I have the asymmetrical polar business figured out. The key is to think of alternating between any two points on the polar, not necessarily on two different tacks.

The attached figure shows two hypothetical speed polars, one symmetrical (blue) and the other asymmetrical (red). Each polar is the locus of points formed by sailing at a steady speed in a given direction relative to the true wind. You might call this the single-point performance, because each point of sail is considered independently.

Now consider what it means to alternate sailing between two different true wind angles, such as A and A'. The line A-A' represents the average performance in a given direction, such as the black arrow, with the assumption that the switch between the two operating points can be done in zero time. If the arrow is closer to A than A', the boat will spend more time sailing at A, and if the course is half way in between then the boat will spend equal time at A and A'. So the line A-A' represents the average performance for the pair A,A'. You can do this for any pair of points on the polar.

The line B-B' represents the average performance of two points that are on opposite tacks. You can slide the points around the polar to try to optimize the performance. The optimum upwind performance will be found when the line is tangent to the polar at C-C'. The boat should sail at C or C' for any course that lies between the two points, such as the second black arrow. The same thing is true for sailing downwind - the best performance would be obtained by drawing a line tangent to the two lobes and alternating jibing between the two tangent points, as shown by D-D'.

The same thing goes for an asymmetrical polar. Where the polar is not convex, draw a straight line tangent to the polar to bridge the non-convex portion and alternate between those points. The upwind optima for the asymmetrical case is E-E'. E and E' are close to, but not the same as, the points with the best Vmg upwind on each tack. E is a little more free and E' is a little closer than you would sail if each lobe was one tack of a symmetrical polar. The difference isn't great, but I think you can see that the line connecting the two best Vmg points would be inside the line shown on the diagram.

The same thing goes for polars that are not convex because of sail changes, such as between working sails and spinnaker. You bridge the lobes of the polars with and without spinnaker, and the two tangent points indicate the courses you should sail for a reach that is too tight for the spinnaker's best performance. You need to sail at the tangent point with jib alone until you get to the layine for the tangent point with spinnaker, and then pop the chute. The average speed over the course will be represented by the straight line connecting them on the polar.

I realize navigators already know this stuff, but it took a while for the penny to drop for me.

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### tspeerSenior Member

So, how does this relate to a nonuniform wind? The choice of reference height will affect the true wind direction and the wind shear and gradient relative to that height. But regardless of the choice of reference height, the boat speed polar relative to the water will be the same. It will, however, become rotated relative to what one is calling the true wind direction. So it can become asymmetrical in the wind axis system even if there is some axis about which it is symmetrical.

To optimize the performance, one would apply the procedure in the post above: make the polar convex and sail the tangent points to obtain a course where the polar does not coincide with its convex hull. This will work regardless of what one chooses for the reference wind direction. So best performance is not necessarily best Vmg!

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### philSweetSenior Member

One side note. You had suggested that a reasonable definition of VMG was a perp to the tangent segment. In general, this is not the slowest point of sailing. That's only true if angle EOE' is a 90 or, if EO and E'O are equal.

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### tspeerSenior Member

True. But it would give the same value for Vmg on both tacks, it's a definition that would be independent of the choice of reference height, and the line would be perpendicular to the true wind direction just like for the ideal wind case (symmetrical polars, no wind shear). The boat speed and true wind angle would be different for the two tacks, but I suppose that can't be helped if the polar is asymmetrical.

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### philSweetSenior Member

Nope. It wouldn't be perpendicular to true wind. That is not something you can monkey with. It is what it is. The VMG would merely be centered on the plot. Think about a bunch of plots all shown this way. Chaos.

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### tspeerSenior Member

Ah, but we monkey with the true wind all the time, depending on where we measure it! Most boats only have one set of wind instruments, at the masthead, which fits very well with the One Watch principle. ("A man with one watch always knows what time it is. A man with two watches is never sure.") But if we have a means of measuring the wind at different heights, the picture is not so simple.

Take a look at the first file I've attached. It shows my hypothetical symmetric polar rotated by +10 deg from the symmetrical baseline. If the true wind direction varied by 20 deg from deck to masthead, this is what you would see depending on what height you chose to mount your wind instruments. You'd get the same boat performance but it would seem to be skewed, depending on where you happened to put your measurements. In reality, the polars would all overlay each other if you plotted them vs course over ground instead of the true wind direction indicated by your instruments. Although these polars all have an axis of symmetry, even a symmetrical polar can look asymmetrical, depending on the reference wind direction.

If, for the purpose of determining Vmg, you picked the direction that was perpendicular to the line tangent to the upwind lobes, then you would get the same true wind direction regardless of what height you chose for your wind instruments. This is shown by the dashed lines.

The second figure shows the same thing, but this time applied to the asymmetrical polar. To my eye, rotating the polar this way makes it look more "natural". Vmg in this orientation is more meaningful because it shows the advancement of the dashed line.

I guess what I'm doing is turning the normal definition of Vmg on its head. The normal definition is "the component of the boat's velocity that is parallel to the true wind." Instead, I'm defining Vmg as being the component of the boat's velocity that is perpendicular to the tangent line. That makes the most sense with regard to optimizing the boat's performance. Then, I use the traditional definition to infer the true wind direction from the Vmg.

In practice, I think this could be applied by sailing on one tack and then the other. Sailors have a pretty good idea when they are sailing near best performance on a given tack. You would then calculate the Vmg direction by taking the arctangent of the difference in north and east velocities on the two tacks.

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### philSweetSenior Member

After writing a lengthy epistle, I finally reallized that at least half the problem is semmantics. I will happily concede that point to you. If True Wind nothing more than a frame of reference ID then my above comment about monkeying with it is not right. But I think most of us regard it as having a tight connection with the real entity as well. If multiple instruments are being used, one must track those as Indicated Wind A and B or some such.

Dealing with an asymmetrical polar is not much of a problem in the current arrangement as far as I'm concerned. Any one who has sailed a lateen or sprit rigged boat is well aware of the issue. I concede that for the rotated symmetrical case there are no issues with what you propose.

My issues pertain to how the AW polars and TW polars are related to one another, how the polars are used. If you first produce a set of AW polars, the data you record will necessarily be with repect to some IAW. Currently, you would apply a transform to produce TW and you are done. I'm not really sure what you have in mind here. You would need to produce a rotated AW plot, bias the wind direction indicator by a equal amount, then generate a TW plot. I think that works.

If you have several wind instruments, you still need to know which one to use. Getting late. Gotta go.

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### ErwanSenior Member

Wind Shear & Coriolis ?

Hi Everybody,

Thanks for putting on the table the asymetric polar issue.

Sorry if my question is a bit candid:

I would like to know if the "wind shear" as mentionned in this thread, is the consequence of the Coriolis effect , while the wing gradient is the consequence of the atmospheric boundary layer on the water?

Best regards

EK

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### tspeerSenior Member

There could be lots of reasons for the wind shear. For example, say the wind is blowing at an angle to the opening of a bay that is enclosed by headlands. The wind may be blowing more directly through the opening in the headlands at low level, but parallel to the high altitude winds at upper levels. Gusts will also produce wind shear over shorter time periods.

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### tspeerSenior Member

That was exactly the problem that got me thinking about these problems. Each of the wind instruments would be equally valid, if corrected for the yacht's motion and position errors. The wind fundamentally does not have a single direction, but the boat's c.g. does have a single velocity.

I now realize that for best performance, Vmg should be defined as the component perpendicular to the line connecting the two operating points on the polar. Progress tangent to the line is made by adjusting the time spent at the two operating points. One seeks to maximize Vmg by choosing the operating points. In the case of the asymmetrical polar, the Vmg one should optimize is not perfectly aligned with the upwind direction. Perhaps it's better to simply recognize that this Vmg is not the same as the Vmg in the windward direction, just as the windward Vmg is not the same as the radial Vmg given by GPS. Sailors know it's misleading to try to maximize the radial Vmg, and it turns out it's also a little misleading to try to maximize the windward Vmg. Windward Vmg, however, is a pretty good first guess, especially for a given instant when the opposite tack performance is not known.

Similarly, the way to handle the non-uniform wind problem is to simply state the height of the reference wind (10m, masthead, etc.) and define gradient and shear from there. There's no right answer for the true wind direction, especially if Vmg is decoupled from the wind direction itself.

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### ErwanSenior Member

I have been searching a little bit, and even if a bit trivial, I found a Wikipedia link addressing the Ekman spiral, quite interesting for me.

Thanks for this smart topĂ®c

Best regards

EK

http://en.wikipedia.org/wiki/Ekman_spiral

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### daiquiriEngineering and Design

It sounds like a feature related to a very local geographical configuration around a given seaway. And what would be typical values of a shear due to these spot-dependent conditions? Whilst I agree that it might be something to consider in the design of a boat which will race in these local conditions only, I am wondering how important are actually these considerations for the design of an all-round boat, which will not be used for just that lake, or just that bay?

Or did I perhaps misunderstood the spirit of this discussion? Perhaps it is not related to the design optimization but rather to establishing an optimum strategy for a given local wind?

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### ErwanSenior Member

Daiquiri,

I had this question in my mind since a long time, but it was not a priority.

If I had this question in mind, it is because, in middle 90 before Olympics in Savanah, a Tornado team: Christophe Clevenot and his crew, told me they were adjusting the sail twist, after tacking, in order to meet the Coriolis effect, As I was a rookie, I stored it my memory until now. My too small experience with A-Cat didn't provide me with any insights regarding this issue.

Regards

EK

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### Mikko BrummerSenior Member

There is often lots (15-30 deg on the height of a 15 m mast) of wind shear early in the spring in Finland, when the water is cold, and much colder than the air (water 6-7 deg, air 15 deg). The surface flow can be all laminar (not even small surface waves), and different tacks require a completely different sail trim.

On many occasions wind shear is a sign of an upcoming windshift - the wind will shift in 15 min. or so to the direction of the tack that is lifted. In other words, the wind at altitude fills down to the surface. One such occasion can be the shifting of the gradient wind to the sea breeze direction - shear is then a very important warning in a race.

I also remember that as youngsters when sailing trapeze dinghies, there were days when it simply paid to point on starboard tack and sail fast and low on port. At the time we did not understand the reason.

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