Are there theories that assert about the ideal shape (i.e., ignoring structural constraints) of a propeller?
Just looking at the shapes that emerge in practice, it appears that both the fluid speed relative to the propeller and the fluid's viscosity are factors that radically affect the optimal shape. Are there any laws that relate these factors?
From what I have observed in practice, I would conclude that a ducted fan with blade angles matched to the desired Mach number, and diameter matched to the desired power, is a maximally efficient propeller, meaning that it converts rotational energy into the maximum possible mass flow to its rear. Is this (or some statement like it) correct?
And if so, does it apply across the full range of viscosity and flow speeds – from hypersonic air flight to trawling under water at just a few knots?