Evaluation of the Biot–Savart Integral for Deformable Elliptical Gaussian Vortex Elements

Abstract

This paper introduces two techniques for approximating the Biot-Savart integral for deforming elliptical Gaussian functions. The primary motivation is to develop a high spatial accuracy vortex method. The first technique is a regular perturbation of the streamfunction in the small parameter $\epsilon = \frac{a-1}{a+1}$, where $a^2$ is the aspect ratio of the basis function. This perturbative technique is suitable for direct interactions. In the far field, this paper studies the applicability of the fast multipole method for deforming elliptical Gaussians since the multipole series are divergent. The noncompact basis functions introduce a new computational length scale that limits the efficiency of the multipole algorithm; however, by imposing a lower bound on the finest mesh size, one can approximate the far-field streamfunction to any specified tolerance.