Journal of Engineering Mathematics 26: 349-361, 1992. © 1992 Kluwer Academic Publishers. Printed in the Netherlands. 349 Inversion of the source and vorticity equations for supercavitating hydrofoils SPYROS A. KINNAS Department of Ocean Engineering, Massachusetts Institute of Technology, Room 5-221, 77 Massachusetts Avenue, Cambridge, MA 02139, U.S.A. Received 19 March 1991; accepted in revised form 15 August 1991 Abstract. The problem of a supercavitating hydrofoil of general shape with leading edge cavity detachment is addressed in linear theory in terms of unknown source and vorticity distributions on the foil and cavity. The related singular integral equations are inverted analytically and the cavitation number, the source and vorticity distributions are expressed in terms of integrals of quantities which depend only on the hydrofoil shape and the cavity length. Numerical algorithms for computing these integrals accurately and efficiently are given. I. Introduction The analysis of the flow around a supercavitating hydrofoil is an essential tool in the overall design of high speed hydrofoil boats. It also constitutes the basis of more complicated flows such as those about supercavitating, ventilated or surface piercing propellers. Linear theory was first applied by Tulin to the problem of supercavitating symmetric sections at zero incidence and zero cavitation number [1], then to the problem of general camber meanlines at zero cavitation number [2], and to a supercavitating flat plate at incidence and arbitrary cavitation numbers [3]. It was subsequently extended to supercavitat- ing hydrofoils of general shape at non-zero cavitation numbers by Wu [4], Geurst [5], Parkin [6], Fabula [7], and Nishiyama and Ota [8]. The previous authors formulated the problem in terms of the complex perturbation velocity function, which was then determined from the application of the boundary condition on the foil, on the cavity, and at infinity. Hanaoka [9] formulated the linearized partial and supercavitating hydrofoil problem in terms of the perturbation velocity potential which he expressed in terms of singularity distributions on the foil. He also gave series representations for the cavitation number, the hydrodynamic coefficients and the cavity shape when the hydrofoil shape could be expressed in terms of polynomials in the chordwise coordinate. More recently, the non-linear flow around supercavitating hydrofoils has been addressed by employing numerical boundary element (panel) methods [10, 11, 12]. These methods discretize the hydrofoil and cavity surface into panels and apply the exact kinematic and dynamic boundary conditions on the exact cavity surface whose shape is determined iteratively. A drawback of the numerical non-linear methods however, especially in three dimensions, is the large computing time which is associated with the computation of the influence coefficients for every new cavity shape in the iterative process. Even though non-linear theories are more accurate, linear theories are more versatile in the design process. In addition, linear theories, especially when applied to supercavitating hydrofoils, provide a very good first approximation in the iterative process for determining the