Using global interpolation to evaluate the Biot-Savart integral for deformable elliptical Gaussian vortex elements

This paper introduces a new method for approximating the Biot-Savart integral for elliptical Gaussian functions using high-order interpolation and compares it to an existing method based on small aspect ratio asymptotics. The new evaluation technique uses polynomials to approximate the kernel corresponding to the integral representation of the streamfunction. We determine the polynomial coefficients by interpolating precomputed values from look-up tables over a wide range of aspect ratios. When implemented in a full nonlinear vortex method, we find that the new technique is almost three times faster and unlike the asymptotic method, provides uniform accuracy over the full range of aspect ratios. As a proof-of-concept for large scale computations, we use the new technique to calculate inviscid axisymmetrization and filamentation of a two-dimensional elliptical fluid vortex. We compare our results with those from a pseudo-spectral computation and from electron vortex experiments, and find good agreement between the three approaches. Key words. Vortex methods, Navier-Stokes equations, Biot-Savart integral, high Reynolds number flow, electron vortices