Vortex detection has been a long-standing and challenging topic in fluid analysis. Recent state-of-the-art extraction and visualization of vortices in unsteady fluid flow employ objective vortex criteria, which makes feature extraction independent of reference frames or observers. However, even objectivity can only guarantee that different observers reach the same conclusions, but not necessarily guarantee that these conclusions are the only physically meaningful or relevant ones. Moreover, a significant challenge is that a single observer is often not sufficient to accurately observe multiple vortices that follow different motions. This thesis presents a novel mathematical framework that represents physically realizable observers as the Lie algebra of the Killing fields on the underlying manifold, together with a software system that enables the exploration and use of an interactively chosen set of observers, resulting in relative velocity fields and objective vortex structures in real-time. Based on our mathematical framework, our system facilitates the objective detection and visualization of vortices relative to well-adapted reference frame motions, while at the same time guaranteeing that these observers are physically realizable. We show how our framework speeds up the exploration of objective vortices in unsteady 2D flow, on planar as well as on spherical domains.

On the tangent bundle of a Riemannian manifold the most natural choice of metric tensor field is the Sasaki metric. This immediately brings up the question of infinitesimal symmetries associated with the inherent geometry of the tangent bundle arising from the Sasaki metric. The elucidation of the form and the classification of the Killing vector fields have already been effected by the Japanese school of Riemannian geometry in the sixties. In this thesis we shall take up the conformal vector fields of the Sasaki metric with the help of relatively advanced techniques.