On the high-accuracy approach to flow simulation aroundthe airfoils by using vortex method

Some problems connected to vortex methods development for 2D incompressible flow simulation around airfoils are discussed. In the numerical schemes and algorithms which are normally used in the vortex methods, the airfoil is approximated by a polygon consists of rectilinear panels, and the the vortex sheet intensity, which is placed on the airfoil surface line and simulates the airfoil's influence the flow, is assumed to be piecewise-constant or piecewise-linear function. The most accurate mathematical models and numerical algorithms, based on such approaches, provide the estimation for the order of the error between O(h) and O(h ²) for the average vortex sheet intensity over the panels (h is the maximal panel length). In the present research a new approach is developed, where the vortex sheet intensity is assumed to be discontinuous piecewise-linear or piecewise-quadratic function and the curvature of the panels is taken into account. In order to compute the coefficients of the main system of the algebraic equations, the least squares method is used instead of commonly used no-through or no-slip conditions at separate control points or over the panels in the average. It is shown, that the developed approach is much more accurate in comparison with previously known ones: for some test problems (flows around a circular airfoil, an elliptical airfoil and Zhukovsky airfoil) it permits to obtain numerical solution which has error of order O(h ⁴). (paper)