Efficient semi-analytical integration of vortex sheet influence in 3D vortex method

The original numerical scheme is developed for vortex sheet intensity computation for 3D incompressible flow simulation using meshless Lagrangian vortex methods. It is based on tangential components of the velocity boundary condition satisfaction on the body surface instead of widespread condition for normal components. For the body triangulated surface the corresponding integral equation is approximated by the system of linear algebraic equations, which dimension is doubled number of triangular panels. Vortex layer intensity on the panels assumed to be piecewise-constant. The coefficients of the matrix are expressed through double integrals over the influ- ence and control panels. When these panels have common edge or common vertex these integrals become improper. In order to compute them it is necessary to exclude the sin- gularities, i.e., to split the integrals into regular and singular parts. Regular parts are expressed by smooth functions, so they can be integrated numerically with high precision by using Gaussian quadrature formulae. For singular parts exact analytical integration formulae are derived. The developed approach allows to raise significantly the accuracy of vortex layer in- tensity computation in vortex method for flow simulation around arbitrary 3D bodies. The test problem of flow simulation around the sphere is considered. The exact analyt- ical solution is known for it, and the developed numerical scheme provides more accurate results in comparison with ‘classical’ 3D vortex method, especially when non-uniform un- structured triangular meshes are used for bodies surface representation. It allows to use arbitrary triangular mesh on body surface and to refine mesh near sharp edges, what is especially important for flow simulation around bodies with complicated geometry