Accurate solution of the boundary integral equation in 2d lagrangian vortex method for flow simulation around curvilinear airfoils

The problem of numerical solution of the boundary integral equation is considered for 2D case. Viscous vortex domains (VVD) method is used for flow simulation, so vorticity is generated on the whole surface line of the airfoil, and there are a lot of vortex elements close to the airfoil. The aim of the research is to provide high accuracy of numer- ical solution of the integral equation; at the same time the computational complexity of the numerical algorithm should be at rather low level. The third-order accuracy numerical scheme, based on piecewise-quadratic solution representation on the curvilinear panels is presented, approximate analytic expressions are obtained for the matrix coefficients. These schemes work perfect in the case of potential flow simulation, when vorticity is absent and also when vortex elements are placed rather far from the airfoil surface line. A trivial way to the accuracy improvement for the closely located vortices, which consists in extremely fine surface line discretization, leads to unacceptably high numerical complexity of the algorithm. This problem is solved by developing semi-analytical correcti- on procedure which makes it possible to achieve high accuracy at extremely coarse surface line discretization. For example, in the model problem of flow simulation around elliptical airfoil with 2:1 semiaxes ratio only 20 panels are required to achieve the error level less than 1 % for arbitrary position of the vortex element in the flow