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I think you gave an excellent explanation of why High Aspect Ratio wings work better than Low Aspect ratio ones. I never thought about the concept of ‘down wash’. I always thought that the high pressure air leaking around the tip spoiled the Low Pressure on top of the wing. therefore one needed to increase the Angle of Attack to increase both the amount of High Pressure under the wing and the Low Pressure on top, to make up for this loss. But of course doing this only increases the amount of tip Vortex, which of course magnifies its effect, which means you may need to further increase the Angle of Attack to make up for that, until, with a low enough Aspect Ratio, the damned thing just won’t fly.
I learned this as a kid, by mounting the wings on a balsa glider so that the span ran length wise. I would then throw it and it would promptly fall to the ground.
But I think keels are a whole other matter. This is because They travel through water, which is for, all practical purposes, non-compressible. This, In my thinking, causes the high pressure side of the Keel to be more important than the low pressure side.
So, in my mind, the keel acts more like a snowplow than a wing, especially when you have Aspect Ratios significantly lower than 1.0. What this means, if I am correct, is that there are diminishing losses, one one end of the Aspect Ration spectrum, as well as diminishing returns on the other.
A good example of this is an experience I once had with my Siren 17. I had no engine at the time, and I had sailed into a cove infested with tree stumps. After quickly realizing my mistake, I decide to get the Hell out of there. But the tight entrance was dead upwind, and I had retracted the swing keel and the deep rudder blade. The rudder blade was still immersed but it trailed behind the rudder stock like a lengthened hockey-stick blade.
At first, I tried to paddle out, but the wind was too strong. It just blew me further into the cove.
Next, I thought of ‘motor sailing’ out, using the the main sail and the paddle. So I raised the main sail, and before I could start paddling, the boat started sailing at some speed. I soon realized the impracticality of my plan. The boat was going to sail faster than I could paddle. So I just let it sail.
After changing tacks several times, I noticed that the entrance to the cove seemed closer. So I kept it up. In less than half an hour, I was at the mouth. About three tacks latter, I was free. So I lowered the swing keel and rudder blade and sailed home.
My boat had just done the seemingly impossible. It had sailed to windward with next to no Lateral Area (the swing keel retracts all the way into the shallow dead rise hull.
The next day, with a brisk wind blowing, I decided to try this again, out in the lake. Sure enough, it did it again. I was doing about 140 degree tacks and was probably making at least ten degrees of leeway, maybe fifteen. But that still meant fifteen to twenty degrees made good to windward.
Now that I think about it, it was probably the very Low Aspect Ratio retracted rudder blade that was doing most of the windward work. The fore foot of the hull was probably holding the bow upwind, so some angle could be cranked into the rudder.
This might explain how Chine Runners work. They hold the hull upwind, so some angle must be cranked into the rudder, to keep the bow out of the eye of the wind. But Matt’s boats have deep, high Aspect ratio rudders. And the are rather large. And if you look a the rig placement, it is usually further back than what is typical.
The Chine Runners do the same job the the fore-foot of my boat did, but because they are further back, they must be larger and more effective. They truly do work as a snow plow, as they cause the water to well up, on the high pressure side. The low pressure side is completely ignored.
The advantage of this system is that there is no ‘board to deal with. the disadvantage is that there is this very deep and large rudder, which must be un-shipped when ever approaching really shallow water. As Matt usually sails in deeper water, this is no problem.
One of the projects on my ‘bucket list’ is to make a model sailboat, which has a double chine, dead flat bottom, so the hull itself has almost zero Effective Lateral Area. Then I would try various keel and rudder configurations on it, with a standard sail. One combination would be a somewhat deep fore-foot fin and a big rudder. But mostly the mission would be to test my two theories on keel Aspect Ratios.
The first of which is: Keel Area = ((((Sail Area + Profile Area)^0.5)+ Keel Length)/29)+(Sail Area* 0.025)
The Profile Area is the entire area of the boat’s profile above the Water Line.
The second is: Pointing Ability = 45 degrees + ((Keel Length/Keel Depth)-1.0), with the 45 degrees meaning how far the boat is pointed OFF the wind.
So, if you have a keel that satisfies my first theory, and it has a Length to Depth ratio of 2.0, It should be able to sail about 47 degrees off the wind. If you have a keel with a Length to Depth ratio of 4.0, you will have to sail 49 degrees off the wind. But if you have a keel with a Length to Depth ratio of 0.25 you should be able to point 44.25 degrees off the wind. Of course modern deep keels have efficient airfoil water lines and act more like wings, so can probably do much better.
As you can see, both of these theories limit how far you can go with a shoal draft keel. Eventually, the keel is so long that it needs so much more area that this additional area increases the required depth. And the pointing ability becomes so diluted, that it is hardly worth the effort. But some times it may be. For instance, suppose you have a row boat which you want to add an auxiliary sail to. And you want it to have some upwind ability. But you don’t want the trouble of carrying a ‘board of one sort or other. So you put a skeg-and-cutwater keel on it, which has to extend just a few inches below the original hull depth. This adds considerable whetted area and surface friction, but it has far less parasitical drag than a permanently extended shorter and deeper one. This strategy was often used in the days of sail. Pea pod row boats were often fitted with this sort of keel, so they could sail when stronger winds made rowing difficult
So far, I have only scant evidence to back up my theories, only the fact that my theory seems to be pretty effective at predicting the size of long keels on existing boats.
The pointing ability theory is just a shot from the hip, but seems to have the ring of truth. I would love to test it, and hope someday I will be able to.
Neither of these theories take into account the profile shape of the keel, so if you have a deep ‘V’ hull, for instance, you can put some of the keel in front of the belly and some of it in back. This would give you the classic ‘skeg-and-cut water’ bottom profile that sailing ships used to have. The only stipulation is that the aft portion must be at least as deep as the forward portion, if not deeper.