David Oettinger | Searcharxiv

arXiv2020

Explicit Unsteady Navier-Stokes Solutions and their Analysis via Local Vortex Criteria

We construct a class of spatially polynomial velocity fields that are exact solutions of the planar unsteady Navier-Stokes equation. These solutions can be used as simple benchmarks for testing numerical methods or verifying the feasibility of flow-feature identification principles. We use examples from the constructed solution family to illustrate deficiencies of streamlines-based feature detection and of the Okubo-Weiss criterion, which is the common two-dimensional version of the broadly used Q-, Delta-, Lambda-2- and Lambda-Ci-criteria for vortex-detection. Our planar polynomial solutions also extend directly to explicit, three-dimensional unsteady Navier-Stokes solutions with a symmetry.

physics.flu-dyn↗

arXiv2016

An Autonomous Dynamical System Captures all LCSs in Three-Dimensional Unsteady Flows

Lagrangian coherent structures (LCSs) are material surfaces that shape finite-time tracer patterns in flows with arbitrary time dependence. Depending on their deformation properties, elliptic and hyperbolic LCSs have been identified from different variational principles, solving different equations. Here we observe that, in three dimensions, initial positions of all variational LCSs are invariant manifolds of the same autonomous dynamical system, generated by the intermediate eigenvector field, $ξ_{2}(x_{0})$, of the Cauchy-Green strain tensor. This $ξ_{2}$-system allows for the detection of LCSs in any unsteady flow by classic methods, such as Poincaré maps, developed for autonomous dynamical systems. As examples, we consider both steady and time-aperiodic flows, and use their dual $ξ_{2}$-system to uncover both hyperbolic and elliptic LCSs from a single computation.

math.DS↗