Efficient solution of Goldstein's equations for propellers with application to rotor induced-power efficiency

Betz and Prandtl (1919) presented the optimum velocity distribution for a rotor in axial flow having an infinite number of blades. Goldstein (1929) derived an expression for the circulation that would give the ideal inflow of Betz-Prandtl. Goldstein offered an elegant, numerical solution to this equation in order to find the optimum circulation to give Betz induced flow. He presented solutions for two blades at a number of inflow ratios and for four blades at one particular inflow ratio. The objective of this work is to develop a more computationally accurate and robust method of finding the optimum circulation for the ideal propeller. We look for a solution that would be taken to any desired accuracy and applied for any number of blades and any tip-speed ratio. With such a solution, one can have benchmarks against which to compare other methodologies. In addition, an accurate solution will allow computation of induced power efficiency for the Goldstein optimum such that other blade designs can be measured against it.