Optimum Actuator-Disk Performance in Hover and Axial Flight by a Compact Momentum Theory with Swirl

Abstract

A new compact form of momentum theory is introduced for actuator disks including swirl. The new form unifies both the axial and angular momentum balances into a single momentum equation, applicable over the entire range of thrust and power coefficients. While completely consistent with earlier momentum theories, such as that of Glauert with swirl, the compact form allows analytic expressions for the parameters of a Betz actuator disk and reveals additional insight into the limiting efficiency of rotors, propellers, and wind turbines. The compact form also allows a completely closed form for the truly optimum Glauert rotor. We will also present results from the Betz hypothesis as practically optimum. Closed-form results presented here include the practically optimum values of induced flow, inflow angle, thrust, induced power, and efficiency. Closed-form expressions are also given for practically optimum twist, chord distribution, and solidity in the presence of profile drag (along with the resulting overall efficiencies). This report also gives a closed-form solution for the truly optimum rotor in hover, based on the Glauert optimality criterion.