On the Calculation of the Vortex Sheet and Point Vortices Effects at Approximate Solution of the Boundary Integral Equation in 2D Vortex Methods of Computational Hydrodynamics

Abstract

When simulating incompressible flow around airfoil using vortex methods of computational hydrodynamics, it is necessary to solve the boundary integral equation with respect to the intensity of the vortex sheet at the airfoil surface line. For its approximate solution, the Galerkin method with a piecewise-constant and piecewise-linear representation of the solution on the airfoil surface line can be applied. The coefficients of the system of linear equations, arising from the procedure of discretization of the integral equation, e xpress the influence of the vortex sheet located on the panels of the airfoil surface line on other panels; the components of the right-hand side vector express the influence of point vortices that simulate a vortex wake in the flow domain. For the case of the airfoil surface line approximation with a polyline consisting of rectilinear panels, exact analytical expressions are obtained for the coefficients of the matrix and the right-hand side vector; for the case of taking into account the curvature of the panels, an approach for their approximate calculation is proposed and all the necessary formulae are derived.