The high-speed submerged hydrofoil in a fluid of finite depth

This paper provides a solution for the steady, two-dimensional, irrotational, free-surface, attached flow of a fluid of finite depth past a fully submerged hydrofoil advancing at constant speed above a rigid horizontal lower boundary in the limit of infinite Froude number. The solution holds for arbitrary far-field fluid depth, foil length and angle of attack, and height of the foil above the lower boundary. The flow domain is considered as the image under a conformal map of a preimage domain in an auxiliary complex plane. Formulae for the complex flow potential, the complex velocity and the derivative of the parameterizing map are constructed in closed-form in terms of the auxiliary complex variable using special functions defined through Schottky groups. The formulae are explicit except for a set of four mapping parameters determined numerically as the roots of a nonlinear system. The map is computed by numerically integrating its derivative. An integral is given for the lift coefficient of the foil. Solutions reveal significant free-surface and ground-effect phenomena. Multiple solutions for identical external parameters are found. The two infinite-depth limits, obtained by removing either the free surface or the lower boundary, are discussed.