T-Schemes for Mathematical Modelling of Vorticity Generation on Smooths Airfoils in Vortex Particle Methods

Abstract

New numerical schemes are proposed for solving the boundary integral equation that arises in CFD vortex particle methods of when simulating a plane flow around smooth airfoils. They are based on considering the 2-nd kind integral equation with respect to vortex sheet intensity with a bounded or absolutely integrable kernel instead of traditionally solved singular integral equations of the 1-st kind with Hilbert-type singularity. To solve it, the Galerkin approach is used. It is shown that when approximating the airfoil boundary with a polygon, it is possible to develop schemes of the 1-st and 2-nd order of accuracy, considering a piecewise-constant or piecewise-linear (discontinuous or continuous) distribution of the solution along the panels. The necessary formulae are presented for calculating the components of the matrix and the right-hand side of the system of linear algebraic equations, that is a discrete analogue of the integral equation. They are suitable for modelling of the vorticity generation when simulating the flow around either single airfoil or system of airfoils, including moving and/or deformable ones. The developed schemes can be used in the framework of the viscous vortex domains method as well as other modifications of vortex particle methods, since they only concern the convective velocities of the flow near the airfoil and are not related to methods for modeling viscous diffusion of vorticity