Author(s)

ILIA K. MARCHEVSKY, SOPHIA R. SERAFIMOVA

Abstract

The algorithm of vortex particle methods of computational hydrodynamics for three-dimensional incompressible flows simulation includes the solution of a boundary integral equation (BIE) that describes vorticity generation on the body surface. Instead of the ‘traditional’ approach, which leads to hyper-singular integral equation, an alternative way allows for considering the BIE with weak singularity, that can be solved with acceptable accuracy by using the Galerkin approach with piecewise-constant solution representation. The kernel of the BIE is the gradient of fundamental solution of the Laplace equation. For the panels with common edge or common vertices high-accurate algorithm is developed, that is based on singularity additive exclusion and its analytical integration. The resulting formulae are written down in such a way to avoid accumulation of roundoff errors. The special cases are considered for which ambiguities arise: limit values of the corresponding terms are calculated analytically and linear expansions are presented in neighbourhood of ‘ambiguous cases’ with respect to small parameters which use allows for achieving high accuracy of repeated integrals computation.

Keywords

vortex particle method, boundary integral equation, Galerkin approach, repeated integral, singularity exclusion, common edge, common vertex, limit value, linear expansion