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It is shown that the time-dependent aerodynamic forces acting on a flapping airfoil in forward flight are functions of both axial and normal reduced frequencies. The axial reduced frequency is based on the chord length, and the normal reduced frequency is based on the plunging amplitude. Furthermore, the time-dependent aerodynamic forces are related to two Fourier coefficients, which are evaluated here from computational results. Correlation equations for these Fourier coefficients are obtained from a large number of grid- and time-step-resolved inviscid computational-fluid-dynamics solutions, conducted over a range of both axial and normal reduced frequencies. The correlation results can be used to predict the thrust, required power, and propulsive efficiency for airfoils in forward flight with sinusoidal pitching and plunging motion. Within the range of parameters typically encountered in the efficient forward flight of birds, results obtained from the correlation equations match the computational-fluid...

Propulsion Theory of Flapping Airfoils: Comparison with Computational Fluid Dynamics