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

## TL;DR

This paper introduces a universal scaling law for the wake transition from drag to thrust in flapping foils, based on the cycle-averaged swept trajectory, providing a unified framework for understanding propulsion mechanisms.

## Contribution

The study replaces amplitude with swept trajectory in the scaling law, revealing a universal $St_{\mathcal{T}} \sim 1$ criterion for wake transition across different flapping modes.

## Key findings

- Wake transition occurs at $St_{\mathcal{T}} \sim 1$ for all tested modes.
- Propulsive jets form when the induced velocity exceeds free stream velocity.
- The new metric links wake development directly to kinematic parameters.

## Abstract

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.