Propulsion Theory of Flapping Airfoils, Comparison with Computational Fluid Dynamics

Abstract

The thrust, required power, and propulsive efficiency of a flapping airfoil as predicted by the well-known Theodorsen model are compared with solutions obtained from gridresolved inviscid computational fluid dynamics.A straight-forward summary of Theodorsen's flapping airfoil model is presented using updated terminology and symbols. This shows that both axial and normal reduced frequencies are of significant importance. The axial reduced frequency is based on the chord length and the normal reduced frequency is based on the plunging amplitude. Computational fluid dynamics solutions are presented over the range of both reduced frequencies typically encountered in the forward flight of birds. It is shown that computational results agree reasonably well with those predicted by Theodorsen's model at low flapping frequencies. An alternate model is also developed, which shows that the time-dependent aerodynamic forces acting on a flapping airfoil can be related to two unknown Fourier coefficients. The computational results are correlated with algebraic relations for these Fourier coefficients, which can be used to predict the thrust, required power, and propulsive efficiency for airfoils with sinusoidal pitching and plunging motion.

Propulsion Theory of Flapping Airfoils: Comparison with Computational Fluid Dynamics

“…This assumption allowed him to evaluate the induced velocity on the airfoil due to the wake vorticity and resulted in a wake model dependent on the axial reduced frequency k [7,9]. However, because the wake model he developed was planar, he neglected the vertical oscillations within the wake due to the amplitude of the vertical oscillation of the airfoil, y A (see Fig.…”

“…However, in the high-frequency limit, Theodorsen's model predicts infinite amplitude oscillations in the aerodynamic forces and moments at finite geometric pitching and plunging angle-of-attack amplitudes, which is clearly unrealistic [9]. For a review of this model as well as a presentation of the model using modern conventional notation, the reader is referred to Hunsaker and Phillips [9].…”