Алгоритм движения точечных вихрей в ограниченной области

Abstract

The problem of calculating the paths of motion of a set of point vortices in an ideal incompressible liquid in a limited area is considered. The area is supposed to be piecewise smooth. The stream function at any given time is presented as a sum: the stream functions of point vortices and the potential of a simple layer, the density of which - the density of vortexes on the border - is required to determine. Regardless of time, the stream function harmoniously extends into the exterior of the area and is identically equal to a constant at the boundary, and, therefore, is a Roben potential. Briefly, the paper describes an algorithm for calculating the Roben potential. The unknown density of vortices is sought in the form of a linear combination of a complete system of potentials, and the coefficients are determined by the values of the Roben potential, as a solution to the inverse problem; determination of the coefficients reduces to solving a system of linear equations with a Gram matrix for a linearly independent system of functions. By the function of stream by tangents determine the trajectories of movements of vortices. The results of computational experiments of the motion of several point vortices in a square are presented. A computer analysis of the movement was carried out: two, four and eight point vortices, calculated the trajectories of their movement and found stationary and periodic, stable and unstable configurations.