Hydrodynamics of an unconstrained self-propelled flexible fin with pure pitching motion

The hydrodynamic performance of an unconstrained self-propelled flexible fin undergoing pure pitching motion, free to move in both the horizontal and lateral directions, is investigated using two-dimensional numerical simulations. A systematic parametric analysis is conducted to examine the effects of pitching frequency, amplitude, and bending stiffness on propulsion characteristics. For a uniformly flexible fin, three distinct propulsion regimes are identified: steady swimming, deflected swimming, and backward swimming, each associated with variations in flapping symmetry and wake structure. In contrast, a rigid fin exhibits only steady swimming across all tested conditions, maintaining symmetric kinematics and coherent wake patterns. Although the flapping amplitudes of the rigid fin's leading and trailing edges are smaller than those of the uniformly flexible fin, the rigid fin consistently achieves higher cruising speed and propulsive efficiency. This superior performance is attributed to a small phase lag between the lateral motions of the leading and trailing edges, which promotes more synchronized deformation and thereby enhances momentum transfer. When a spatially non-uniform bending stiffness is introduced into the flexible fin, performance enhancement is achieved through increased flapping amplitudes and improved phase coordination. The resulting deformation facilitates the constructive merging of vortices shed from the leading and trailing edges, producing stronger and more coherent wake structures. Consequently, the non-uniformly flexible fin outperforms not only the uniformly flexible fin but also the rigid fin in overall propulsive performance. Although this superiority is consistently observed in the low-Re regime, the efficiency advantage shifts in favor of the rigid fin at higher Re.