Theoretical optimum rotor performance in hover and climb

Recent results show that Finite-State methods can be used to compute induced power for a rotor in climb. A formal optimization with finite-state models was successfully performed in closed form and that optimization recovered the form of solution for all of the classical theories for a lifting rotor under lightly-loaded conditions. That is, it agrees with the Prandtl solution for an actuator disk with a finite number of blades, with the Betz distribution for a lifting rotor with an infinite number of blades, and with the Goldstein solution for a lifting rotor with a finite number of blades. According to Betz, the optimum induced flow distribution is one that maintains a constant helix screw angle, φ. The induced optimum flow and circulation are nearly identical to that which can be obtained from momentum theory with the Prandtl tip correction term, k. The theory of Betz also applies to the case of hover. The power and thrust can be determined from the lift and drag perpendicular and parallel to the vortex sheet, and can be expressed in terms of the lift and drag coefficients respectively. For the optimum rotor including profile drag, one should find the optimum circulation that will result in the maximum lift-to-drag ratio. If the effect of drag in the vertical direction is neglected, then one can formulate some simple, closed-form results for the lift, power, and thrust in hover of an optimum rotor.