A mathematical model and analytical solution for a self-propelled heaving hydrofoil in close proximity to free surface

The study of self-propelled heaving hydrofoils in close proximity to the free surface is of great significance in marine propulsion, energy harvesting, and underwater vehicle design. However, the complex unsteady flow phenomena and the free surface interactions present challenges in modeling and optimization. This paper proposed a comprehensive mathematical model and analytical solution for a self-propelled heaving hydrofoil operating in close proximity to the free surface. The key hydrodynamic factors, including unsteady lift, induced drag, and their dependence on parameters such as the Strouhal number, aspect ratio, and relative submergence depth, are investigated. Analytical expressions for lift, drag, and thrust coefficients are derived, providing a predictive framework for hydrofoil performance under various operating conditions. The vorticity distribution near the hydrofoil is analyzed using a mirror vortex method and modified pressure distributions to capture the free surface effects. Results reveal that at small submergence depths, the vorticity distribution forms compact, symmetric vortex rings near the hydrofoil surface. As the relative submergence depth increases, the vorticity splits into distinct symmetric peaks, with contour lines becoming smoother and more uniform. Similarly, as the angle of attack increases from 5° to 30°, the vorticity gradient intensifies in the central region, forming high-vorticity closed-ring structures and expanding the distribution range. Analytical results are also compared with the numerical results, validating both the mathematical model and numerical simulations. The main theoretical contributions include the development of a modified Prandtl lifting-line equation that accounts for free surface effects and the derivation of closed-form solutions for unsteady lift and drag coefficients. These findings provide valuable insights for optimizing hydrofoil performance in practical engineering applications.