Asymptotic Theory of Broadband Rotor Thrust, Part I: Manipulations of Flow Probabilities for a High Number of Blades

Modern computer predictions of the frequency spectrum of net thrust for a rotor in turbulent flow display a number of features that call for interpretation. The success of the asymptotic development described here in providing the needed explanations lies in the relative generality of the model. The rotor is fully three dimensional. It is not a cutaway blade row. The theory’s only requirement is that the number of blades be large (six is enough). The most practical aspect of the new work is that it supplies a highly accurate analytical solution for the broadband “haystacking” in a propulsor of high solidity. An important intermediate result is that the probabilistic amplitude of random gusts perceived by that propulsor climbs radially outwards along its blades, solely by virtue of the three-dimensional kinematics of the rotor’s aerodynamic sampling of the turbulent flowfield: The behavior is r³ for the normal-to-rotor flow statistics and r⁴ for the cross correlation of normal and in-plane downwash components. The final effect is a spanwise distribution of sectional loads that is highly concentrated at the blade tips. The turbulence in the present study is homogeneous and isotropic but its integral scale may take on essentially any value and the asymptotic analysis still holds.