Sailing Performance In a Nonuniform Wind

Tom

If you take vmg to mean the speed in the direction your potential for progress is minimized, as in your red line perpendicular to the segment that is tangent to the polar, I think you are good to go. Then, vmg is unique regardless of reference.

The stickiness comes from here -

That's not a deduction, it's a definition of what we are calling a polar. That being so -

isn't consistent with the definition. Its ALL in there. Everything that impacts performance potential is in there. Whether or not we have such polars is another matter. I think few do.

 

I agree. That's why Stan Honey's comment about estimating the speed on the opposite tack is so relevant.

It turns out you can't optimize the tack you're on unless you know something about the opposite tack. If you're trying to find, say, E', the error in E' if you base Vmg on C-E' is a second order effect, compared to basing Vmg on E-E'. So you don't need to know the opposite tack to high precision. But you do need to know, for example, that the performance at E is less than the performance at E' so you can bias your course in that direction instead of sailing at C'.

Another thing that's interesting is if you draw the perpendicular bisector to E-E', it doesn't pass through the origin. The Vmg vector is tilted more away from the true wind direction than is a vector passing from the origin through the midpoint on E-E'. As a practical matter, this means that if you sail on both tacks and then average the heading of the two tacks, that heading will be biased in the direction of best Vmg, even though it will fall short of the Vmg direction. So even with an asymmetrical polar, it's a good first guess as to what direction to optimize. It will be better than optimizing a given tack in the direction of the true wind.

In real life, we never know "best". All we can hope for is to move in the direction of "better". Averaging the headings of the two tacks is better.

 

Is that a fundamental property of all polars, or is it typical of some causes of asymmetry, but not others?

 

I don't think it is a fundamental property. I think you can conceive of quite reasonable situations where the deviation in Vmg direction would be the same as the deviation in average heading.

 

It seems that what is being considered is the movement of the line and the assumption is the movement is perpendicular to it.

However if you consider points on the line you get the diagram below. The numbers represent % time on starboard.

That is, mapping C-C' to E-E'

For simplicity I have made the horizontal components of C and C' equal so that % time on both tacks is equal at the vertical.

Reading off the intersection of E-E' with the vertical the best progress directly to a windward mark would appear to be 42% starboard and 58% port.

 

Yes, that's correct. If you were sailing to a different mark, there would be a different percentage, of course. But that's a good illustration of how much the sailing might be biased to one tack or the other compared to optimizing each tack individually.

The key thing is by sailing E-E' instead of C-C' you do get an improvement in progress to windward. So when searching for E or E', one wants to maximize the velocity component perpendicular to the line, not the velocity component in the direction of the wind.

 

Quoting from post 8

I think I understand what is happening.

Re the bold underlined sentence. If we look at this from a different perspective, the helmsman's only control is over the heading, so try varying that, changes in speed are secondary effects.

I have designed an analogue computer where the headings on starboard and port can be varied and the resultant change in Vmg shown by a rod coming out of the top.

Trying changes in starboard and port headings alternately, each time optimising Vmg. Eventually the mechanism will converge on the maximum possible Vmg.

When you take the cover off, the yellow curved slots represent the polar for the boat.

The blue bar can only move vertically and rotate about its centre, it is incapable of moving perpendicular to itself when it is inclined. The blue bar does arrive at the position where it is tangent to the lobes of the polar.

 

Just realised you don't need to do that. If you pull the measuring stick upwards, when it stops, the bar will be tangential to the polar lobes and you can read the pointers to give the headings.

 

An easier way is to just throw a loop of string around the polar. Draw it tight, and you've formed the convex hull. The string will lie along the polar where the polar is convex, and bridge any hollows with straight line segments.

The bar approach is a good way to visualize optimizing a given tack. Fix the opposite end of the bar at approximately the best Vmg point, and rotate the bar upward for the tack of interest. Then fix that tack, and repeat for the opposite side. That's what you'd be doing when sailing, as you search for best performance on each tack.