Fluid-Elastic Interactions in Flutter And Flapping Wing Propulsion

Abstract

This study seeks to understand the interplay of vorticity and elasto-dynamics that forms the basis for a fluttering flag and flapping wing propulsion, and factors that distinguish one from the other. The fluid dynamics is assumed two dimensional and incompressible, and comprises potential and viscous flow simulations. The elastic solid is one dimensional and governed by the Bernoulli-Euler flexure model. The fluid and elastic solid models are coupled using a predictor-corrector algorithm. Flutter of a flag or foil is associated with drag and we show that the pressure on the foil is predominantly circulatory in origin. The circulatory pressure generated on the foil depends primarily on the slope and curvature. The wake vorticity exhibits a wide range of behavior starting from a Kelvin-Helmholtz type instability to a von Kármán wake. Potential flow simulations do not capture the wake accurately both at high and low mass ratios. This is reflected in the flutter boundary and pressure over the foil when compared with viscous flow simulations. Thrust due to heaving of a flexible foil shows maxima at a set of discrete frequencies that coincide with the frequencies at which the flapping velocity of the foil tip is a maximum. The propulsive efficiency shows maxima at a set of discrete frequencies that are close but distinct from the thrust maxima set of frequencies. These discrete frequencies are close to the natural frequencies of vibration of a cantilevered foil vibrating in vacuum. At low frequencies thrust is a consequence of a strong leading edge vortex developed over the foil and it remains attached to the foil as it is convected due to the favorable pressure gradient presented by the time and spatially varying shape of the foil. At moderate and high frequencies of oscillation the pressure, and consequently the thrust, generated by the foil is non-circulatory in origin and they are high where the accelerations of the foil are high. At high frequencies the leading edge vortex is weak. Except in the low frequency range, potential flow simulations qualitatively compares well with viscous flow predictions. We show that thrust and drag on a flexible foil oscillating in a flow is caused by the phase difference between the slope of the foil and the fluid pressure on it. Propulsive efficiency though is governed by the phase difference between foil velocity and fluid pressure and inertia forces. Thus, the interplay of vorticity and elasto-dynamics determine the behavior of a flutter and propulsion of a flexible foil in a fluid flow.