Axisymmetric Vortex Sheet Motion: Accurate Evaluation of the Principal Value Integral

Abstract

This paper concerns the accurate evaluation of the principal value integral governing axisymmetric vortex sheet motion. Previous quadrature rules for this integral lose accuracy near the axis of symmetry. An approximation by de Bernadinis and Moore (dBM) that converges pointwise at the rate of O(h3 ) has maximal errors near the axis that are O(h). As a result, the discretization error is not smooth. It contains high wavenumber frequencies that make it difficult to resolve the vortex sheet motion. This paper explains the reason for the degeneracy near the axis and proposes a modified quadrature rule that is uniformly O(h3 ). The results are based on an analytic approximation of the integrand, whose integral can be precomputed. The modification is implemented at negligible additional cost per timestep. As an example, it is applied to compute the evolution of an initially spherical vortex sheet.