Circulation Flow around a Thin Plate

Abstract

The problem of plane-parallel flow around of contuer by flow of an ideal incompressible fluid is considered. The current function in general is represented in the form of three terms: the first term is responsible for the flow in infinity, the second term is the simple layer potential and it is responsible for the potential flow, and finally, the third term is the Roben’s potential and is responsible for the circulation flow around the contuer. To calculate the current function of the general circulation flow problem, it is required to determine the densities of two potentials. A simple algorithm for calculating unknown densities is proposed, based on the harmonic continuation of the stream function into the region bounding by the contuer. For approximation, we use special complete systems of basis potentials -- shift systems for the fundamental solution of the Laplace equation with shifts within the region. The approximation coefficients are determined by solving a system of linear equations with a Gram matrix. The algorithm for solving the Roben’s problem is briefly described. The results of computational experiments for a thin plate at different circulation values are presented. The results of computer modeling are compared with a picture of the real flow from the album of M. Van-Dyke. The proposed algorithm can be effectively used to calculate the streamflow function, not only thin, but also piecewise-smooth contours of various geometries.