Study on the Karman gait kinematics of an airfoil in an asymmetrical vortex street

To understand fish swimming behavior in unsteady flows, this paper introduces the Kármán gait model to numerically investigate the hydrodynamics of fish-like swimming in an asymmetric vortex environment, specifically the P + S mode (a pair of vortices are shed from one side of the cylinder and a single vortex from the other side during one oscillation period) created by an oscillating cylinder. The immersed boundary method is employed to model both the fish-like airfoil and the vibrating cylinder. Through simulations across a broad range of controlling parameters, we analyze the advancement efficiency of the airfoil in the P + S mode, the force coefficients, Fourier spectra of hydrodynamic forces, and the interactions between the airfoil and vortices. Our findings reveal that the fundamental phase Φ0 is crucial, as it directly influences the airfoil's position relative to the vortex and affects the forces exerted. Other parameters play a secondary role, primarily reinforcing the effect of the fundamental phase on airfoil–vortex interactions. Furthermore, the vortex pair boosting effect, unique to the P + S mode, enhances the airfoil's thrust and swimming efficiency. The wake environment behind the airfoil is also vital for maximizing benefits from the P + S mode. When the fundamental mode fs, indicative of the airfoil's ability to extract energy from vortices, dominates the Fourier spectra of hydrodynamic forces, it supports the airfoil's motion in the P + S mode. Conversely, when the first harmonic mode 2fs dominates the drag spectrum, it hinders propulsion by reducing the airfoil's thrust in the swimming direction.