Unified Scaling for Flapping Fins

Conventional scaling for steady-fin or flapping-fin forces in a uniform flow is by definition incompatible with the scaling that has been developed for hovering flapping fins. Most flapping-fin-based vehicles will need to operate in all three operating regimes: fixed fin in cruise, flapping in cruise, and flapping in hover. A unified normalization scheme is proposed to enable smooth and optimal transition of the scaling rules between these regimes. A rigid flapping foil was experimentally tested in a towing tank. The generated side force was found to be a function only of the fin bias angle and the vector magnitude of the inflow and flapping speeds. The thrust was found to be a function of the ratio of fin flapping speed to total speed magnitude in addition to bias and speed magnitude. As an application of this scaling, an algorithm to synthesize the foil motion parameters for a desired thrust and side force regardless of vehicle speed was tested experimentally on a single foil in real time.