Flow of an ideal fluid with free surfaces in doubly connected and triply connected regions

Abstract

In the framework of the Tulin model of supercavitating flow, the problem of reconstructing the free surface of a channel and the shapes of the cavities behind two hydrofoils placed in an ideal fluid is solved in closed form. The conformal map that transforms a parametric plane with three cuts along the real axis into the triple-connected flow domain is found by quadratures. The use of the theory of Riemann surfaces (the Schottky doubles) enables the non-linear model problem to be reduced to two separate Riemann–Hilbert problems on a hyperelliptic surface of genus two. The solution to the first problem is a rational function with certain zeros and poles on a Riemann surface. The second problem is solved in terms of singular integrals with the Weierstrass kernel. The essential singularities of the solution at the infinite points of the surface due to a pole of the kernel are removed by solving a real analogue of the Jacobi inversion problem on the surface. The unknown parameters of the conformal map are recovered from a system of certain algebraic and transcendental equations.