Gust Response and Mitigation Through Passive Pitching*

The pitching motion around an elastic joint enables natural wings to keep steady flight by passively responding to gusts. To understand the underlying principles, we investigate a two-dimensional foil free to pitch in a uniform stream. An external pitch moment proportional to the pitch angle is applied to the foil, representing the effect of a torsional spring. We first demonstrate analytically that the quasi-steady variation of any force component, e.g. the lift, can be cancelled by setting the pitching axis through any point along along a straight line that depends only on the geometry and the Reynolds number. Then, we investigate numerically the transient response of a NACA0012 foil to a streamwise gust. We consider initial chord-based Reynolds number values of 1000, 50k and 1M; a solid-to-fluid density ratio of 5; a torsional spring preload of 30 degrees; and a spring stiffness such that the initial angle of attack is 5 degrees. The incompressible Navier-Stokes equations for a Newtonian fluid are weakly coupled with a rotational harmonic oscillator within OpenFOAM. For a fast gust with a duration shorter than a convective length, and with an amplitude of the fluctuation of the same order of the mean flow (e.g. doubling the free stream velocity), the transient forces are governed by the generation of added mass vorticity and their shedding. We find that the amplitude of the lift fluctuations decreases asymptotically with the distance of the pitching axis from the foil, and are already reduced by an order of magnitude when the distance is about one chord length. The kinematics of the pitching foil is well predicted by solving the Newton-Euler equation of motion considering quasi-steady circulatory and added mass forces. Additionally, we find that vortex shedding triggered by the gust at high Reynolds numbers can be suppressed by passive pitching. These results provide new insights into the mechanism by which unsteady loads are generated and on the technology by which they might be mitigated.