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On the high-accuracy approach to flow simulation aroundthe airfoils by using vortex method K S Kuzmina et al Abstract. We consider the methodology of numerical schemes development for twodimensional vortex method. We describe two different approaches to deriving integral equation for unknown vortex sheet intensity. We simulate the velocity of the surface line of an airfoil as the influence of attached vortex and source sheets. We consider a polygonal approximation of the airfoil and assume intensity distributions of free and attached vortex sheets and attached source sheet to be approximated with piecewise constant or piecewise linear (continuous or discontinuous) functions. We describe several specific numerical schemes that provide different accuracy and have a different computational cost. The study shows that a Galerkin-type approach to solving boundary integral equation requires computing several integrals and double integrals over the panels. We obtain exact analytical formulae for all the necessary integrals, which makes it possible to raise significantly the accuracy of vortex sheet intensity computation and improve the quality of velocity and vorticity field representation, especially in proximity to the surface line of the airfoil. All the formulae are written down in the invariant form and depend only on the geometric relationship between the positions of the beginnings and ends of the panels.

The problem of 2D incompressible flow simulation around airfoils using vortex methods is considered. An exact solution for the boundary integral equation with respect to a free vortex sheet intensity at the airfoil surface line that arises in such problems is obtained. The exact solution is constructed for flows around elliptical and Zhukovsky airfoils using the theory of complex potentials and conformal mappings technique. It is possible to take into account the influence of singularities in the flow domain — point vortices which simulate vortex wake. The obtained exact solutions can be used to verify and estimate the accuracy of numerical schemes for the boundary integral equation solution: such procedure is also described in details.