Generic Variational Spacetime Optimization of Vortex Core Manifolds

The detection of vortex structures in fluid flow is a crucial task in continuum mechanics and flow visualization. However, vortex detection is an extremely challenging task that, despite its importance, is not yet fully solved for 3D unsteady flow fields, due to the complexities introduced by the time-dependence of unsteady flow. We introduce a generic variational framework for the computation of optimal vortex cores in 3D unsteady flow that combines a geometric vortex core model with explicit reference frame optimization. Instead of focusing on a specific vortex detection criterion, we use a generically defined Lagrangian that can incorporate different vortex criteria in a unified way. A key insight of our framework is that the two-manifolds comprising 3D vortex cores in spacetime can be obtained by solving the Euler-Lagrange equations in a single time step with only one independent variable. This is enabled by a Lagrangian that is pre-integrated in time according to the pushforward of the underlying flow. The combination of temporal pre-integration and solving for the optimal two-manifold using only one spatial parameter results in an extremely efficient algorithm.