| The dynamic wake model is applied to the optimal propeller systems originally studied by the classic aerodynamicists: Betz, Prandtl and Goldstein. Several modified forms of the model are theoretically developed to extend the applicable range to flight conditions with a large swirl velocity component. Dynamic wake model calculations accurately predict the inflow behavior for helicopter rotors, including axial flow for large tip-speed ratios, (OR/V infinity) ≥ 20. The swirl velocity is a prominent component for small tip-speed ratios (≤5), typical of forward flight for tiltrotor craft such as the V-22 Osprey and the BA609. Dynamic wake calculation results are compared to the closed-form solutions by Prandtl and Goldstein. The exact and approximate solutions correlate strongly for infinite blade cases and finite blade cases with a large tip-speed ratio. The original form of the He-Peters and Morillo-Peters dynamic wake models converge poorly for small tip-speed ratios, due to neglect of the swirl velocity. Derivations are presented for several adaptations of the models to account for the large apparent mass at the inboard blade region. A best modified form is chosen and the associated empirical factor is optimized to correlate well with Prandtl's solution. Error norms for the original and modified forms of the dynamic wake model are presented for propellers of various number of blades and a range of tip-speed ratios. The Goldstein solution is also studied in depth and conclusions are drawn for improving the dynamic wake model. |
