Vortex filament calculations by analytical/numerical matching with comparison to other methods
The calculation of fluid velocity from the Biot-Savart law integrated over vortex filaments has traditionally been computationally expensive. Discretizing the filaments into N vortex elements results in order N2 elemental velocity evaluations per time step. Further, the elemental resolution has been governed by the need to resolve the large velocity gradients in the near field of the filaments, resulting in unnecessarily high element densities in the far field, where the velocities are slowly varying. The method of Analytical/Numerical Matching (ANM) improves the efficiency of the filament velocity calculation without loss of near-field accuracy. This is done by using a far field comprised of computationally inexpensive vortex particles with a large core size for smoothing. The near field is done by an analytical correction which uses a thin physically correct core size to predict the large rapidly varying near-field velocities, and a second correction with the large core size to cancel the local vortex particle error and match to the far-field solution. As such, the ANM method is similar to the method of matched asymptotic expansions. The entire approach has been analytically linearized, which provides additional efficiency and allows unique solution opportunities. Examples are given which illustrate the efficiency and accuracy of the ANM method in vortex dynamics calculations.
