Inviscid scaling laws of a self-propelled pitching airfoil

Abstract

Inviscid computational results are presented on a self-propelled virtual body combined with an airfoil undergoing pitch oscillations about its leading-edge. The scaling trends of the time-averaged thrust forces are shown to be predicted accurately by Garrick's theory. However, the scaling of the time-averaged power for finite amplitude motions is shown to deviate from the theory. Novel time-averaged power scalings are presented that account for a contribution from added-mass forces, from the large-amplitude separating shear layer at the trailing-edge, and from the proximity of the trailing-edge vortex. Scaling laws for the self-propelled speed, efficiency and cost of transport ($CoT$) are subsequently derived. Using these scaling relations the self-propelled metrics can be predicted to within 5% of their full-scale values by using parameters known a priori. The relations may be used to drastically speed-up the design phase of bio-inspired propulsion systems by offering a direct link between design parameters and the expected $CoT$. The scaling relations also offer one of the first mechanistic rationales for the scaling of the energetics of self-propelled swimming. Specifically, the cost of transport is shown to scale predominately with the added mass power. This suggests that the $CoT$ of organisms or vehicles using unsteady propulsion will scale with their mass as $CoT \propto m^{-1/3}$, which is indeed shown to be consistent with existing biological data.