The Algorithm of the Vortex Sheet Intensity Determining in 3D Incompressible Flow Simulation around a Body

Abstract

An original algorithm is developed for vortex methods of computational fluid dynamics for determining the intensity of the vortex sheet on the surface of a body in the flow of an incompressible medium. Unlike the common in the vortex methods approach to satisfying the no-slip boundary condition on a streamlined surface, which is based on ensuring that the normal velocity component of the medium is zero, the proposed procedure is based on a mathematically equivalent condition of equality to zero of the tangent velocity component on the body surface. The unknown intensity of the vortex sheet is assumed to be piecewise constant on triangular panels that approximate the surface of the body. The resulting integral equation is approximated by a system of linear algebraic equations, which dimension is twice the number of panels. The coefficients of the system matrix are expressed through double integrals over the panels. An algorithm is proposed for calculating these integrals for the case of neighboring panels, when these integrals are improper. An additive singularity exclusion is performed and analytical expressions for the integrals of them are obtained. The smooth parts of integrands are integrated numerically using Gaussian quadrature formulae. The proposed algorithm makes it possible to improve significantly the accuracy of the vortex sheet intensity reconstruction when flow simulating around complex-shaped bodies by using vortex methods for arbitrary triangular surface meshes, including essentially non-uniform and having cells with high aspect ratio.