Journal of Engineering Mathematics 19 (1985) 329-339. © 1985 Martinus Nijhoff Publishers, Dordrecht. Printed in The Netherlands. A numerical method for non-linear flow about a submerged hydrofoil L.K. FORBES * Department of Mathematics, Kansas State University, Manhattan, KS 66506, USA (Received June 4, 1985) Summa~ A numerical method is presented for computing two-dimensional potential flow about a wing with a cusped trailing edge immersed beneath the free surface of a running stream of infinite depth. The full non-linear boundary conditions are retained at the free surface of the fluid, and the conditions on the hydrofoil are also stated exactly. The problem is solved numerically using integral-equation techniques combined with Newton's method. Surface profiles and the pressure distribution on the body are shown for different body geometries. 1. Introduction This paper is concerned with the flow of an ideal fluid about a hydrofoil immersed beneath the free surface. The fluid is of infinite depth and flows steadily from left to right, The hydrofoil is assumed to possess a blunt nose and a cusped trailing edge. Linearized theories may be developed by regarding the hydrofoil thickness as a small parameter. This approach is summarized by Wehausen and Laitone ([1], page 583). As in the case of thin-wing aerofoil theory, an integral equation is obtained for the strengths of the vortices distributed along the centre-line of the foil; however, unlike classical aerofoil theory, there is no simple closed-form solution to this equation. Nevertheless, it is still possible to demonstrate that the application of the Kutta condition at the trailing edge gives a bounded fluid velocity there, but an infinite velocity at the leading edge. A numerical investigation of the non-linear potential flow about a hydrofoil has been undertaken by Salvesen and von Kerczek [2]. They solved Laplace's equation in a fluid of fixed finite depth using finite differences; clearly, such an approach is not directly available in the conceptually simpler case of infinite depth considered here. In addition, finite-difference methods are obviously difficult to apply in irregularly-shaped computa- tional domains. However, the authors are able to claim reasonable agreement with experimental data. Further numerical techniques for linearized and non-linear free-surface problems are reviewed by Yeung [3], and the review article by Acosta [4] surveys the general field of hydrofoil vehicles. * Presently on leave at: Department of Mathematics, University of Queensland, St. Lucia 4067, Queensland, Australia. 329