A third approximation to Glauert's momentum theory

Glauert derived a set of momentum equations. He was unable to solve this set but offered two approximations whereby (for either case) he was able to solve for rotor performance. The first approximation is to assume constant circulation-which implies uniform flow. His second approximation is to neglect the effect of wake rotation in the far wake-which results in a nonuniform wake that expands to atmospheric pressure. Either approximation eliminates the possibility of finding the effect of wake rotation in the far wake. We offer a more general, third approximation to Glauert's equations that allows solution of the downstream variables. A change of variable introduces a wake contraction function for which there is a well-defined differential equation. This differential equation is solved numerically to give far-wake behavior for general climb rate and thrust in either the helicopter or the wind turbine state.