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 the deficiencies of streamline-based feature detection and those 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.