Optimization of the motion of a flapping airfoil using sensitivity functions

The motion of a flapping NACA0012 airfoil is optimized by means of numerical simulations for a Reynolds number equal to 1100. The control parameters are the amplitudes and the phase angles of the flapping motion in addition to the mean angle of attack. Sensitivity functions are used to compute the gradient of a cost functional related to the propulsive efficiency of the airfoil and a quasi-Newton method is adopted to drive the control parameters towards their optimal values. The ability of a flapping airfoil to produce sufficient lift and thrust forces for appropriate kinematics is demonstrated. Furthermore, a linear dependence between heaving and pitching amplitudes is found for optimal configurations leading to a constant value of the maximum effective angle of attack roughly equal to 11°. This value corresponds to the angle yielding the maximal lift-to-drag ratio for this Reynolds number when the NACA0012 airfoil does not flap. Previous results such as the high propulsive efficiency when a 90° phase angle exists between heaving and pitching, or the reversal of the von Karman street for a Strouhal number close to 0.2, are confirmed here with a new methodology. Finally, optimal kinematics for various types of missions are given and the corresponding flows are described.