Loads and propulsive efficiency of a flexible airfoil performing sinusoidal deformations

This paper presents the application of state-space airloads theory to a flexible airfoil performing sinusoidal deformations at high Reynolds numbers. Given the two-dimensional motion of a flexible airfoil, we derived the closed forms for the propulsive force, lift force, generalized forces of pitching and bending as functions of reduced frequency k, dimensionless wavelength z, and dimensionless amplitude A/(2b). We also calculate the power required to perform this sinusoidal deformation and the propulsive efficiency. Our results show a positive, time-averaged propulsive force for all k>k₀=π/z, which is when the wave speed is greater than the moving speed. At k=k₀, which is when the moving speed reaches the wave speed, the motion reaches a steady-state with all forces being zero. When k<k₀, the system is the case of energy extraction in which the drag force (negative propulsive force) and wake are causing the airfoil to vibrate. For the propulsive case, the propulsive efficiency decreases from 1.0 to 0.5 as k goes to ∞, or k₀ goes to 0. If there were no wake, the propulsive force would be zero at wavelengths of z=0.569 and z=1.3 for all k, and local optimum at z=0.82. Though these extrema of propulsive force with wavelength are smoothed out by the wake effect, one can still see around z=1.3 (k=2.4) the slope transitions of all three powers in Fig. 9. When k<2.4, the cost for high propulsion become more expensive as more power input is used by wake, thus less efficiency.