Universal scaling law for drag-to-thrust wake transition in flapping foils

• Published in: Journal of fluid mechanics Vol. 872
• Main Authors: Lagopoulos, N. S., Weymouth, G. D., Ganapathisubramani, B.
• Format: Journal Article
• Language: English
• Published: Cambridge Cambridge University Press 10.08.2019
• Subjects: Drag; Efficiency; Equilibrium; Flapping; Fluid mechanics; Foils (structural shapes); Heaving; Kinematics; Reynolds number; Scaling; Scaling laws; Self-similarity; Simulation; Velocity; Vortices
• ISSN: 0022-1120; 1469-7645
• DOI: 10.1017/jfm.2019.361

Reversed von Kármán streets are responsible for a velocity surplus in the wake of flapping foils, indicating the onset of thrust generation. However, the wake pattern cannot be predicted based solely on the flapping peak-to-peak amplitude $A$ and frequency $f$ because the transition also depends sensitively on other details of the kinematics. In this work we replace $A$ with the cycle-averaged swept trajectory ${\mathcal{T}}$ of the foil chordline. Two-dimensional simulations are performed for pure heave, pure pitch and a variety of heave-to-pitch coupling. In a phase space of dimensionless ${\mathcal{T}}-f$ we show that the drag-to-thrust wake transition of all tested modes occurs for a modified Strouhal $St_{{\mathcal{T}}}\rightarrow 1$ . Physically, the product ${\mathcal{T}}f$ expresses the induced velocity of the foil and indicates that propulsive jets occur when this velocity exceeds  $U_{\infty }$ . The new metric offers a unique insight into the thrust-producing strategies of biological swimmers and flyers alike, as it directly connects the wake development to the chosen kinematics, enabling a self-similar characterisation of flapping foil propulsion.