Faster upwind tacking by taking turns towing?

Discussion in 'Sailboats' started by h.t.hall, Mar 24, 2005.

  1. h.t.hall
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    h.t.hall Landlubber

    Hi, everyone. Is it possible for two identical sailing vessels, connected by a long rope, to tack into the wind faster, collectively, than either of them could do on its own?

    I think I can safely claim to be more ignorant about sailing than almost anyone else on this forum: I've actually never set foot on a sailboat. (And this after having lived both in the San Francisco Bay Area and on the shores of Lake Geneva, quel dommage!) As a result, anything I say is likely to be complete nonsense in terms of practical sailing, and it may be difficult to even understand what nonsense I intend since I'm not familiar with the correct terminology.

    I'm a mathematician (here's where most of you stop reading), and just for fun the past couple of days I've been thinking a bit about what makes it possible to sail against the wind. In an extremely simplified mathematical model, tacking is possible because the force vector of the wind is projected three times. (To project a vector means to separate out the useful part of it from the part that's just pushing sideways.) First the wind vector is projected in a direction perpendicular to the sail, then the force from the sail to the boat is projected parallel to the keel, and then the fact of constantly switching tacks amounts to a third projection: when the keel isn't pointed in the direction you want to go, your motion has forward and sideways components, and only the forward component is useful.

    This is where the mathematics comes into play. It's impossible to project a vector directly to a negative multiple of itself (there is no "upward component" to "down"), nor is it possible to do so in two steps. It is possible, however, to project a vector three times and wind up with a (shorter) vector which points directly opposite to the original vector. A bit of calculus shows that the best you can do, in terms of getting as long an opposite vector as possible, is to make all three projections change the angle by 60 degrees, which each time will reduce the length of the vector by a factor of two. In the (ridiculously simplified) sailboat model, this corresponds to having the sail at 30 degrees to the wind (that is, the perpendicular force of the sail on the mast should be 60 degrees away from the direction the wind is pushing), and then having the ship (and keel) pointed into the wind and 60 degrees away, the sail thus splitting the angle. For an abstract yacht, a direct headwind is only 1/8 as effective as a direct tailwind. Does that sound remotely accurate for real yachts?

    Having worked that out for myself (as has surely been done hundreds of times before), I wondered: What if there were a way to project the vector through an additional intermediate stage? A projection which changes the direction by 45 degrees reduces the length by a factor of the square root of two; four such projections would yield an opposite-pointing vector 1/4 as long as the original rather than 1/8 as long. (In fact, the greater the number of independent projections, the better. If you could manage 180 ideal projections, each deflecting the direction of the vector by only one degree, you could retain 97% of the force to apply in the opposite direction. Think of a marble shooting around a bend.) In the open sea, I doubt such a thing is possible: each additional projection requires something new to push against, something besides the wind and the boat and the sea. But what if there were two such boats, pushing--or rather pulling--against each other?

    Here's the idea: we have two identical ships, both of them trying to head north against a north wind. They are connected to each other by means of a (very long, very strong) rope. Ship A is far to the east and slightly ahead of ship B, and it takes a nearly crosswind heading so as to be partially towing ship B (the optimal bearing in the theoretical framework is close to 68 degrees, says the math, with the sail splitting the angle between that and the wind). Ship B now has the benefit of an eastward-pointing force in addition to whatever comes from its sails, and it points itself more forward than it normally would, almost as if it were tacking east into a north-northwest wind rather than into a north wind. (The optimal theoretical bearing for ship B is close to 32 degrees, with the sails still splitting the difference between that and due north.) Although ship A is using the wind more effectively, it is making less forward progress, and gradually ship B will pull ahead of ship A (but still far to the west of it). Once ship B is as far ahead of A as A was originally ahead of B, they trade roles, with ship B taking a heading of 292 degrees and partiallly towing ship A, which takes a heading of 328 degrees.

    Making all the ridiculous assumptions that allow one to think of the problem mathematically rather than in practical terms, the average forward thrust of the two ships against the wind should be nearly .178 of what they'd have from a tailwind, rather than the factor of .125 they could each optimally expect if untethered. In other words, the two idealized ships cooperating can do 42% better than they could accomplish separately. (A complete fourth projection would give the equivalent of .25 tailwinds, representing an improvement of 100% over the .125 we get with ordinary tacking. So that's obviously not quite what's happening here--this scheme allows each boat to send the wind through an average of something like three-and-two-fifths projections, whatever that means.)

    My question is this: Does this have the remotest bearing on the real world of sailing? I've ignored all sorts of--well, I've ignored just about everything. The towing cable needs to be long enough to remain mostly east-west while ship B gains on ship A, which means very long indeed, considering that frequent changes of tack would completely nullify the marginal advantage of the mutual towing scheme. It needs to be strong enough to tow a ship--a ship that's being dragged at an angle, no less--and yet light enough not to drag in the water (which would devastate any improved efficiency). I'm not sure, either, where the cable should to be fastened to each ship. (To the mast? Anywhere else, and it would probably have to be continually unhooked and reattached elsewhere with each change of tack--as if there wasn't enough to do already while changing tack.) I can't begin to imagine how such a cable should be rigged without getting in the way of everything else, and the new source of asymmetric strains on the ship itself are quite different from what sailcraft are usually designed for, I would guess.

    I'd be curious to know whether any of you have ever encountered something resembling this setup, and to try to hash out whether the theoretical marginal advantage could ever be worth the additional hassle and material requirements of practice. I seem to recall something about fishing trawlers sometimes working in pairs, dragging the net between them as they went, for example--in the days before steam, did they find it advantageous, when tacking starboard against the wind with a net between them, for the starboard ship to take a more crosswind bearing and fall behind, only to catch up on the tack to port?

    If anyone's really really interested, I can pull out the relevant equations (haven't double-checked them, hope I haven't goofed). Happy sailing!
     
  2. SailDesign
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    SailDesign Old Phart! Stay upwind..

    Would that it were that simple :)
     
  3. h.t.hall
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    h.t.hall Landlubber

    Yeah--I've done about a zeroth-order approximation here, whereas those of you who really care about these questions are doing fourth- or fifth-order approximations, I'm sure, whatever the computer can handle. In particular, I haven't even accounted for things like the difference between actual wind and apparent wind. (Which is a perfectly valid assumption if the speed of the craft is negligible compared to the speed of the wind--but let's hope that's not the case!)

    I am curious, though, whether the essential idea survives a slightly more realistic analysis--the question being: Suppose you're allowed to "borrow" and then symmetrically "repay" an additional lateral force (which is essentially the purpose the tow cable serves). Does this permit, at least in principle, a more efficient conversion of backwards-blowing wind to forward-moving thrust?
     
  4. usa2
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    usa2 Senior Member

    i dont think it would work, because if you have a boat that is capable of going 8 knots upwind and you tie it to an identical boat, its not like its going to accelerate the boat. In order for them to collectively go faster, than one boat would have to be quite a bit faster so it started towing the other one. Or, maybe if you attached them together rigidly, they might possibly go faster. who knows..
     
  5. Skippy
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    Skippy Senior Member

    h.t.hall: I'm a mathematician (here's where most of you stop reading) ...

    Actually, the point where I stopped reading is when I scrolled down to find an entire page of nonstop text. :) Now that I've saved the page, finished my other internet reading, and made a cup of coffee,

    h.t.hall: ... the fact of constantly switching tacks amounts to a third projection ...

    ??? Oh, okay. You're saying the tacks cancel out over time. You don't mean instantaneously.

    Just a practical note, the sail is not directly projecting an existing vector. It's generating its own vector (force) from a relative motion (momentum). The magnitude of the force (the sail's lift) will depend on the angle of attack, which needs to be kept relatively small (<10 degrees??) when you're going upwind to avoid stalling. That means that to maximize the magnitude of the sail's force, the first projection will be 80 degrees or more.

    h.t.hall: I'd be curious to know whether any of you have ever encountered something resembling this setup ...
    My guess would be No.

    h.t.hall: Here's the idea: we have two identical ships, both of them trying to head north against a north wind. They are connected to each other by means of a (very long, very strong) rope....

    The trailing boat will always be going faster than the lead boat, so both ships' rigs will have to be trimmed on each tack.
    Here's another point: Each boat makes better progress when trailing, but little or none when towing. So what makes you expect to see a net advantage? The .178 factor applies only half of the time to either boat. .178 / 2 == .089 < .125, right? To express it another way, with the two boats on the same tack, you're not really taking advantage of the tack/projection phenomenon.

    h.t.hall: I seem to recall something about fishing trawlers sometimes working in pairs, dragging the net between them as they went, ... did they find it advantageous, when tacking starboard against the wind with a net between them, for the starboard ship to take a more crosswind bearing and fall behind, only to catch up on the tack to port?

    That might be a practical solution to the heavier load on the lead boat.
    Edit: Okay, never mind that. The windward boat is being dragged leeward by the net, and the leeward boat is being dragged windward. The leeward boat will have to tow the windward boat just to compensate for that, so I would expect them to stay on the same course.

    h.t.hall: In particular, I haven't even accounted for things like the difference between actual wind and apparent wind. (Which is a perfectly valid assumption if the speed of the craft is negligible compared to the speed of the wind--but let's hope that's not the case!)

    So we'll assume this applies mostly or only to large heavy ships.

    Here's another approach, probably of comparable realism:

    First, you would construct the Mother of all crossbeams, very long, and attach it between the boats directly across the wind. Place the cargo on a separate parasitic vessel attached to the midpoint of the beam. Keep the two sailboats very light to minimize drag. Now start the boats facing away from each other at the midpoint of the beam, and sail them on opposite tacks away from each other, sliding along the beam across the wind while remaining attached to it and pulling it (and the cargo) into the wind. The biggest source of water drag is the cargo vessel, which moves directly upwind at a slower speed, so it's more efficient. When the boats reach the ends of the beam, they both tack and head back toward the midpoint. This ignores the weight and windage of the crossbeam.
     
  6. SeaDrive
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    SeaDrive Senior Member

    I have not drawn any diagrams, but I think I have have a glimmer of what you are suggesting. Two boats sailing tied together, sailing almost parallel, their bows on a line perpendicular to the wind. The leeward boat bears off slightly, generating more power and gives the windward boat a bit of a tow. The windward boat takes advantage of the tow to point a little higher. Yes?

    1) The boats have to stay the same distance apart, but you have the courses diverging.

    2) The windward boat has to generate more lift from the keel, hence more drag.

    3) The boats will have tack often, resulting in less power over time.


    Not to mention the practical factors of communication, drag of the connecting rope in the water, etc.

    Interesting thought, though.
     
  7. asathor
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    asathor Senior Member

    Servo amplifiers......

    Just like small servo amplifiers can be used to correct the output of a large beefy amplifier you can imagine a small boat that points well pulling on the bowspit of a large boat (rotating it into the wind) - the problem however, is that the sum of the work doesn't change. The little boat will not make any progress on its own thus paying for the large boats progress.

    100kw/h=100kw/h

    In case I am wrong (happened before) I attached a polar diagram for an excellent performing boat the Borresen BB10 (it kills "modern" boats when the going gets rough).

    If you can borrow Deep Blue for a couple of days you should have an answer.
     

    Attached Files:

  8. water addict
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    water addict Naval Architect

    No it does not.
     
  9. SuperPiper
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    SuperPiper Men With Little Boats . .

    HT Hall:

    You need to draw a cartoon to show what you mean.

    I have no clue what you have discovered. But, I wonder if it can be applied to a catamaran with a mast on each hull?
     
  10. yipster
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    yipster designer

    only half getting it yet, first thought i have would be tacking will at some point take the wind out of the sail of the other...
     
  11. gonzo
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    gonzo Senior Member

    NO. There would be induced drag from the rope. Also, the different movement and speeds ( pitching, rolling, yawing) would make the boats pull from each other. This produced more turbulence around the mast and sails which slows the boats down.
     
  12. Skippy
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    Skippy Senior Member

    I think he has the towing boat on a beam reach, and the trailing boat pointing very high, maybe even 30 degrees or so. The trailing boat kind of gets sling-shotted into the wind, then they both tack and the roles are reversed.
     
  13. asathor
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    asathor Senior Member

    Thanks Skippy..

    SLINGSHOT!

    With a rigid teeter the whole thing will undoubted cancel out or get worse for the reasons suggested above by several contributors.

    But if you can store the energy! Like a slink shot, you might have something.

    What you need is to transfer energy from the boat on the more efficient point of sail to the boat on the less efficient point of sail.

    The windward boat thus needs to be the one that is being towed at first - when it has gotten as far as possible it bears off and the leeward boat rounds up to point higher - in fact it probably needs to tack, crosses its course and becomes the windward boat. In anything remotely resembling the real world that won't work with a rigid teeter because the boat on the wide tack will be "getting out of reach" due to its higher speed.

    We therefore have to invent the "Skippy Cord_tm:cool:" the 1 mile lossless bungee cord with 100% rebound.

    Now you have something. I still think you need to rent Big Blue by the time the variables get in there to determine which courses this would work at.
     
  14. gonzo
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    gonzo Senior Member

    Also, as every racer knows, course changes slow down the boat. The turbulence on the rudder among other things take a lot of energy.
     

  15. SailDesign
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    SailDesign Old Phart! Stay upwind..

    One word, and I think we can put this post to bed:

    ENTROPY.

    Thank you - G'Night!

    :)

    Steve
     
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