Asymmetric traveling wave undulation in a two-dimensional foil: Hydrodynamic performance and wake structure

Enhancing the performance of biomimetic underwater vehicles requires a deeper understanding of the fluid dynamics of fish-like undulation kinematics. This study numerically investigates the hydrodynamic performance and wake structures of a two-dimensional (2D) foil undergoing asymmetric (outward vs retract) traveling wave undulation. A kinematic model integrating quarter-specific asymmetry (parameterized by ζ) with a foil of constant midline length is used to systematically explore the effects of Reynolds number (Re=250–2000), Strouhal number (St=0.1–1.0), and motion asymmetry (ζ=0.6–1.4). The results show that asymmetric motion with a slow-outward/fast-retract stroke (ζ>1) improves thrust generation and shifts the drag-thrust boundary to lower values of Re and St, although at the expense of reduced efficiency compared to symmetric motion (ζ=1). We propose a generalized thrust model for the 2D undulating foil, C¯T=c1Rec2St3+C¯T0, which is valid within the studied parameter space. The coefficients c1 and c2 exhibit a strong dependence on ζ. Furthermore, four distinct wake structures are identified, showing transitions governed by Re, St, and ζ. While hook-shaped vortices (ζ>1) enhance thrust and tadpole-shaped vortices (ζ<1) impair it, both are linked to low propulsive efficiency due to power dissipation from lateral motions. The direction of wake deflection can be altered by vortex interactions and can be determined from the distribution of the dimensionless cycle-averaged velocity and pressure, along with their relative magnitudes on the upper and lower surfaces of the foil. These findings may help provide insights into the flow physics of fish-like propulsion and offer practical implications for the design and control of bio-inspired underwater vehicles.