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We have conventional foils that appear to work up to 50 knots at least. We know that with supercavitation work is being done and in principle this work can be harnessed to do the work of a foil.
I'm not sure whether a supercavitating foil is capable of working similarly to a "slower foil" ie: producing side-lift... AFAICT, the research is simply focusing on reducing skin-friction.
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So is the problem that conventional foils WONT work with supercavitation, or is it a problem of dealing with a transition state like going supersonic in air?
Anything could work "in supercavitation mode" - just like sticking an Ares rocket engine to your Volvo would make it go supersonic - any shape will supercavitate, but not very well...
As you hinted, there is a problem when transitioning from "starting-cavitation" to stable supercavitation - basically the drag increases dramatically during this phase, thus requiring a suitable increase in thrust.
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What causes the unstable tip vortex? Is it due to the sharp edge and the pressure difference between the sides?
yep - well... maybe... Thinking about it, the tip instability might be able to be modeled using XFoil...
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If that occurs at 20 knots I wonder what's happening at the leading edge of our conventional fins. I know there is a general requirement to keep the leading edge somewhat rounded. Could that be the reason?
The problem seems to be that at high speeds you have supercavitation whether you like it or not so how do you handle it. Is that correct?
Well... I was a little hopeful of that "20 knots" figure...

I think a sharp leading edge
might work - although I am very skeptical... maybe a squared-off bit right on the edge...
At very high speeds, yes you always have supercavitation - but without a rocket motor, your probably not going to come anywhere near it with conventional engines.
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So back to the supercavitating foil - You have a sharp leading edge generating an oscillating vortex about the low pressure side. You say you need 100 knots to make the bubble stable. What's the problem with it being small but unstable? Does it cause vibration or damage the fin? If the chord is short enough would this help?
100 knots is minimum value that has shown in tank results (plus high speed photography), to produce a stable bubble around the object. As you said, the sharp tip is simply there to cause a pressure variation at a much lower speed - whether that pressure change is of the good kind is a separate question... and you would need to bubble on both sides of the foil, not just the low pressure side.
An unstable bubble will collapse earlier in the flow, thus causing water to touch the foil - increasing drag very dramatically (aka shock waves, turbulent water, etc). Vibration and damage is probably a minor problem compared to the thrust issue...

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Why do you need a high angle of attack? Is that just to keep the back of the fin away from the collapsing bubble?
Not sure what is being asked... the aft-part of foil shape? then yes, ie: simply to ensure that there isn't a negative relative-pressure gradient.
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If the foil is asymmetrical that implies that the low pressure side is not going to be used as a high pressure face so you are free to fiddle with it.
Note sure about asymmetry... since there is pretty much no knowledge of side-lift when supercavitating, we cant even assume there is a "low pressure side"... although I do suspect there is, eg:
Here I am trying to show what happens when we change the angle of attack... normal flow should cause the bubble to eventually collapse somewhere aft of the foil. When the angle changes, there would be change in pressure from each side, giving us some control (these are awful pics - the fluid line shouldn't touch...).
I also suspect that the tail of the bubble might elongate somewhat - which may cause the equivalent of directional thrust in a direction off-axis to the direction of travel.