Comparison of dynamic wake models with closed-form optimum propeller solutions

The following article explains how to extend the He-Peters dynamic wake model to be applicable to propellers 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 ((ΩR/V_∞) ≥ 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. Results for dynamic wake calculations are compared to the classic analytic solutions for optimally efficient propellers 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 He-Peters dynamic wake model converges poorly, for small tip-speed ratios, due to neglect of the swirl velocity at the inboard blade region. A derivation is presented for adapting the mass matrix, with an empirical factor, to account for the nonzero mass at the blade root. Results from the modified dynamic wake model correlate well with the two-dimensional exact solution, based on a chosen optimal value for the empirical factor. The error norm for the modified He-Peters model, versus Prandtl's solution, remains constant at approximately 3% for tip-speed ratio range, 5 ≤ μ₀ ≤ 20. The error norm diverges to large values in a concerted manner for μ₀ < 5. The applicable range of the He-Peters dynamic wake model for axial flow is greatly expanded by substituting the modified mass matrix into the calculation equations. An optimal empirical value can also be chosen for convergence with Goldstein's solution at the medial and outboard blade regions, but further research is required to include the far inboard vortex effects present in the three-dimensional exact solution.