Assessment of two vortex formulations for computing forces of a flapping foil at high Reynolds numbers

Abstract

Several vortex formulations for computing the lift and drag/thrust forces on a flapping foil are compared in terms of simplicity, accuracy and suitability for using with experimental data of mea- sured velocity and vorticity fields. These methods are very useful to physically understand the aerodynamic performance of the flapping foil, and therefore the mechanisms of animal swimming and flight. In particular, we consider the case of a two-dimensional heaving foil at high Reynolds numbers and, using the results of accurate numerical simulations, we compare the performance of three force formulations based on the vortical impulse and Lamb’s vector, projection of Lamb’s vec- tor, and wake integrals. We find that the formulation based on the vortical impulse is appropriate when all the vortices generated by the flapping foil remain close to the foil, such as in a starting motion or in hovering, but may become quite inaccurate for analyzing forward flight or swimming. In these later cases the projection method is much more appropriate because it only takes into account vorticity close to the moving solid body at each instant. In fact, we find that, for high Reynolds numbers, the performance of the projection method is always better, even in hovering, due to its simplicity and because it neatly separates the contribution of the added-mass force from the circulatory or vortical force component. Finally, it is shown that the standard integral momen- tum method, which is quite simple to implement in a forward motion using experimental data of the velocity field in the wake behind the solid body, is much less accurate than the methods based on vortex force formulas.