The accuracy and robustness of the algorithm are first accessed by numerical tests based on a synthetic vortex ring. The radii and circulations varying along the centerlines are estimated based on the vortex atoms converged on the vortex centerlines. By iteratively updating the positions of these vortex atoms, the algorithm manipulates them into marching toward the underlying vortex centerlines. The novel aspect of this algorithm is to represent the vortex field as a collection of discrete vortex atoms. The algorithm not only extracts the vortex centerlines but also returns the vortex radii and circulations varying along the centerlines. This article proposes an integrative algorithm for visualizing and quantifying the vortices in three-dimensional flows. Numerous investigations aiming to reveal the underlying physics behind complex flows highlighted the important roles of vortices.
