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

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 chord-line. 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 \(St_{\mathcal{T}}\sim 1\). Physically the product \(\mathcal{T}\cdot f\) 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.