Acoustic Power Minimization in Hovering Propeller Design

Classical rotor theory states that a uniform induced velocity distribution along the span minimizes the mechanical power required to generate a prescribed thrust. The present work extends this result by also considering the acoustic emission. Using a multipole expansion approach, explicit expressions for the acoustic power emitted by a hovering propeller are presented without resorting to far-field pressure integrations. Each blade is modeled as a lifting line, and the acoustic field is decomposed into lift, drag, and thickness contributions. After showing that accurate results can be obtained even when considering only the first term of the series, the optimum problem is formulated. The objective function is defined as a weighted sum of the mechanical and acoustic power of the rotor. By means of Lagrange multipliers, the problem is reduced to a nonlinear system, which is solved numerically for different values of the weighting factor, tip Mach number, and thrust coefficient. The largest relative noise reductions occur for low thrust coefficients and high tip Mach numbers. For a two-bladed propeller, tonal noise is reduced by up to 44% with moderate mechanical-power penalties. Among the secondary parameters, the blade number was found to have the most significant influence on the achievable tradeoff.