The simplest design was used 100 years ago. Straight blades with slight tilt allow the wheel to rotate with minimal leading edge drag.
Simple yes, most efficient no, according to Poncelet and other engineers. My view is that in this application, the more blade area the better. Curved blades have more area. If manufacturing would have been easier, I would have made decreasing radius curved blades to better utilize the water that is being washed over the tops of the blades. I'm looking at the possibility of lengthening the blades another foot, as well as going 20% wider on the lengthened portion to maximize efficiency.
For efficiency you want:
* Curved blades
* Long enough that the water doesn't go over the inner edge, but falls back in the same passage that it entered.
A continuously tightening curve is nice, but not critical.
The idea is that:
* The water enters essentially straight on to the blade, attaching rather than creating turbulence.
* It rises up the curve, using part of its kinetic energy to push against the curved blade, the rest to make it rise up against gravity, trading kinetic energy to potential.
* Once it's reached its closest to the axle height, it falls back down, again along the curved blade. This gives the blade a further push while partially suspending the water on its way down. As the potential energy of height is again converted to kinetic energy of motion (in this case, mostly downward), much of it is transferred to the wheel and little is left in the falling water by the time it reaches the surface.
So the water enters the wheel turbulence free, pushes the retreating wheel forward on its way up and again on its way down, and leaves the wheel moving downward as slowly as practical and forward at about the rate of the wheel's motion, and at a height slightly higher than on entry (which you can think of as either "Bernoulli says slower is higher pressure, pushing the water up" or "the water's moving more slowly than when it entered so it needs a larger cross-section to achieve the same mass flow rate".
* The kinetic energy in the forward motion is the "Betz limit" mandate of "you have to leave some energy in the fluid to get it out of the way of new fluid bringing additional energy".
* The higher exit is a loss, as some of the water fights its way uphill to leave. This is also part of the Betz limit "Energy left in the fluid to get it out of the way", corresponding to the spreading of the widening of the streamlines of the slowed wind downstream of a mill.
* Turbulence and friction are also losses, though these can be made small.
So a smooth curve, with the entrance tangent to the incoming water, an exit moving at the mill's spin rate, and the water staying between the blades and flowing smoothly, is a high efficiency device, because the water leaves with little energy and most of the difference between that and the input energy is converted to shaft horsepower rather than turbulence, friction, or other losses.