Science paper //The International Summer Scientific School High Hydrodynamics, June 2002, Cheboksary, Russia. P.331-338.
This paper presents a solution for compressible flow around two circle contours with a pair of symmetric vortices based upon the one for incompressible flow obtained by the author. The solution for incompressible flow around two circle contours in the presence a pair of symmetric vortices is based on the potential theory, which uses the theory of complex variable function (TCVF). The TCVF makes it possible to write the infinite convergent series for complex potential flow from dipoles and vortices. Formulation of the infinite convergent series is based on the multi-stage use of the theorem about circle in the form given by Milne-Thomson. This solution can be considered as an exact solution for potential flow around two circle contours in the presence a pair of symmetric vortices. In this paper, solution for compressible flow is based on the Burago approximate method, which uses hypothesis of "stabilization of the streamlines". Here it is shown that Burago approximation agrees very well with the Khristianovich theory. The results presented here indicate that for thick airfoils at high Mach numbers Burago approximation has better agreement with experimental data than calculations on the Karman-Tsien formula, while the Prandtl-Glauert approximation has a large inaccuracy. Examples of calculations for incompressible and compressible flows around two circle contours with various locations and circulation of the symmetric vortices are considered. One can note that the range of Mach numbers of the compressible flow where the present theory can be applied is narrower for bluff body than for the thin airfoil, because the flow rarefaction on a surface of such bodies is greater.
