The high-speed submerged hydrofoil

This paper gives, in the limit of large Froude number, a closed-form, analytical solution for steady, two-dimensional, inviscid, free-surface attached flow over a submerged planar hydrofoil for arbitrary angles of attack and depths of submergence. The doubly connected flow domain is conformally mapped to a concentric annulus in an auxiliary plane. The complex flow potential and its derivative, the complex velocity, are obtained in the auxiliary plane by considering their form at known special points in the flow, and the required conformal mapping is determined by explicit integration. The four real solution parameters are determined as the simultaneous roots of four real nonlinear algebraic equations arising from the flow normalisation. The explicit form allows accurate evaluation of various flow quantities, including the lift on the foil, and these are related to the large-Froude-number results in recent numerical solutions.