Scaling laws for 2D pitching hydrofoils

Modeling the forces and torques on 2-D pitching airfoils is critical for understanding the locomotion of fish and birds. Traditional linear theories can predict some of the forces, but break down at larger amplitudes or frequencies. We propose modifications that account for large-amplitude shear layers and velocities induced by trailing-edge vortices. The scalings suggest cost of transport should scale with mass to the -1/3 power, which is consistent with our inviscid simulations and biological data. (This work was done in collaboration with the Biofluids Research Lab at Lehigh University.)


Authors: Keith Moored, Daniel Quinn

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 bioinspired 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 ~ m^(−1∕3), which is indeed shown to be consistent with existing biological data.

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