Condition of boundary integral equations arising from flow computations

Abstract

Stating flow boundary conditions for normal and tangential velocities leads to Fredholm integral equations of the first and second kind. Approximating the vortex distribution on pressure and suction side by piecewise linear functions with equidistant knots, collocation yields a commonly overdetermined system of linear equations with the vortex densities as unknowns. There are several possibilities for collocation. The question arises what kind of integral equation and what collocation points are to be preferred. Our main conclusions based on condition numbers of the resulting linear systems are: - The equation of the first kind with Cauchy singular kernel does not appear to be worse conditioned than the equation of the second kind with regular kernel. - Collocation points for the equation of the first kind are better placed in between the knots, for the equation of the second kind at the knots. - The circulation is mainly determined by the equation of the first kind, the mean velocity level by the equation of the second kind. Employing simultaneously equations of the first and second kind generally improves the solution and allows to cope with aerodynamic profiles of any thickness and loading.