Some questions were posed via PM that I'd like to respond to on the thread (I've already responded by PM as well)...
First I had to make sense of what a propeller even does, and from what I gather it produces propulsion by pulling the vehicle into the air in front of it.
That's essentially correct. It's basically a couple of wings swung around an axis. The lift generated pulls the vehicle forward.
The wheel/propeller are connected through some sort of transmission.
That's correct. There is a fixed ratio transmission between the propeller and wheels in the downwind case, or between the turbine and wheels in the upwind case.
In both cases we use ratchets to let the prop or turbine spin freely if it gets ahead of the wheels. This relates to practical matters and allows us to prove we're not using stored energy. Ratchets are not a necessary part of the design.
Next the gearing is set so that the propeller has to spin faster than the wheels turn (not too sure of the exact ratio here but I think it needs to be greater than 2).
Not exactly. It's less about the rotational rate of the wheels and propeller, and more about the rate each advances through (or on) its medium. For the downwind cart, the prop needs to advance through the air more slowly than the wheels advance over the ground. It's exactly the opposite for the upwind vehicle.
In the initial accelerating phase i.e. when the cart is slower than the wind you are simply drawing power from the wind until you reach some cutoff point.
In the downwind case, the whole vehicle (prop included) is simply a bluff body being pushed by the wind initially. As it begins to move, the prop starts to turn (since it's geared to the wheels). This of course begins to produce thrust. There's a smooth transition between the bluff body phase and the thrust phase. Prop thrust is the sole contributor by the time we reach wind speed.
In the upwind case, the turbine is turned by the wind, and that in turn turns the wheels. There is no change in phase in this case.
Once you hit this cutoff point the cart then actually begins to draw power from its own forward movement...
Yes and no. We have to consider the relative wind over the vehicle when doing the analysis, but we never use any stored energy or momentum of the vehicle to our advantage.
Whether a wind powered vehicle can "make its own wind" is a bit more of a philosophical question, but my inclination (without going into a lot of detail) is to say that it really can't. You could argue either side of this, but you don't get something for nothing as it might otherwise imply.
...and the potential difference between the velocity of the wind at the propeller and the wind speed at ground level (this part is probably the most confusing aspect of the whole machine, and I am not entirely sure how this process works).
It sounds like you're talking about wind gradient. While it's possible to design a vehicle that exploits wind gradient, this one does not. This vehicle exploits the energy available from the relative velocity of the two media (ground and air).
