Analytical modelling for a three-dimensional hydrofoil with winglets operating beneath a free surface

The current study focuses on establishing a theoretical lifting surface model for predicting the hydrodynamic loads acting on the three-dimensional hydrofoil with winglets, which is considerably influenced by the proximity to the free surface through finding the three-dimensional Green’s function for the planar and vertical horseshoe vortices operating below a free surface. The hydrofoil surface is decomposed into a finite number of elements along the span direction and the chord directions, each of which can then be represented by a horseshoe vortex. The linearized free surface boundary condition is applied to analyze the influence of the free surface on the hydrofoil as well as the winglets. The thickness problem is considered using the source distribution among the hydrofoil and winglets surfaces and the analytical Green’s function that satisfies the linearized free surface boundary condition is used. As a sample application, numerical examples were conducted to show the performance of the hydrodynamic characteristics for the hydrofoil with winglets as a function of the Froude number. It was concluded that there are significant efficiency benefits from using winglets inside the free surface proximity effect. These results are substantiated by the comparison with the available published data.