Hierarchy of Numerical Schemes for the Bie Solution in 2d Flow Simulation Using Vortex Methods

Abstract

The problem of 2D incompressible flow simulation around airfoils with sharp edges and corner points is considered. The solution of the boundary integral equation with respect to vortex sheet intensity, arising in Lagrangian vortex method, has weak singularity that cannot be resolved correctly in the framework of the existing Galerkin-type numerical schemes. Known numerical schemes with piecewise-constant or piecewise-linear numerical solution representation provide solution reconstruction with high quality and the second order of accuracy only for piecewise-smooth bounded solutions. For singular solutions their order of accuracy goes down to the first. It is shown that wrong behaviour of numerical solution takes place only on the panels that adjust to the corner point. Modified numerical scheme is developed that is based on the Galerkin-Petrov approach and allows us to obtain integral characteristics of solution (the components of the added masses tensor) with the second order of accuracy. The scheme can be easily implemented in codes developed for flow simulations using the vortex particle method.