Inviscid scaling laws of a self-propelled pitching airfoil

TL;DR

This paper presents inviscid scaling laws for a self-propelled pitching airfoil, accurately predicting thrust and speed, and introduces novel power scaling laws that incorporate added-mass and vortex effects, aiding bio-inspired propulsion design.

Contribution

It develops new scaling laws for self-propelled airfoils that include added-mass and vortex effects, improving prediction accuracy and aiding propulsion system design.

Findings

Thrust scaling matches Garrick's theory.

Power scaling deviates from classical theory at finite amplitudes.

Cost of transport scales with organism mass as CoT ∝ m^{-1/3}.

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.