Reversed von K\'arm\'an 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.
Categories: physics.flu-dyn