  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.
