Mitigating the sound of a flapping airfoil using optimal structural properties distributions

Abstract

We study the effects of structural properties distributions on the sound radiation of a thin elastic airfoil. Focusing on flapping-flight in a high-Reynolds and low-Mach regime, we seek optimal conditions to mitigate flapping-sound while maintaining aerodynamic efficiency. The airfoil is immersed in a two-dimensional potential flow field, and is subjected to leading-edge actuation in the form of a small-amplitude periodic heaving motion. The near-field dynamics is governed by Euler-Bernoulli's beam equation of motion and analyzed using thin airfoil theory in conjunction with a discrete-vortex wake model. The near-field results are introduced as an effective dipole-type source term to the right-hand-side of the Powell-Howe acoustic wave equation, and the associated far-field sound is calculated using Green's function formulation. Considering the elastic flapping configuration, we seek optimal material properties and a linear thickness distribution to reduce the sound of an otherwise rigid airfoil while retaining the lift amplitude of the rigid configuration. To this end, the aeroacoustic model is introduced into an optimization scheme, where minimal sound amplitude is sought in the lift direction, subjected to the aforementioned lift force constraint with a 33% assigned tolerance value. The relatively wide tolerance produces in turn an effective Pareto front, reflecting the trade-off between the competing objectives of lower sound levels and aerodynamic efficiency. Compared with the rigid heaving airfoil, over 10 [dB] sound reduction was obtained for the optimal flexible configuration producing the same lift amplitude. The Pareto front also displays a linear relation between sound reduction ratio and lift amplitude ratio, indicating a further substantial sound reduction of up to 30 [dB] for smaller lift ratio values. The effect of optimal structure thickness distribution on the equi-lift sound reduction mechanism is two-fold: The motion and wake dipoles are shifted to an antiphased mode thus reducing the total sound signal, and the motion dipole which represents the unsteady forces, exerted by the airfoil on the fluid while transversing, is fixed in magnitude, thus retaining the lift amplitude value. The system dynamics leading to the opposing sound dipoles is evoked by phase-locking the airfoil motion and circulation, and following Kelvin's theorem, antiphase-locking the airfoil motion and wake circulation. Implications on flapping flags and streamers is briefly discussed as well.