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-, Δ-, λ2-, and λci-criteria for vortex-detection. Our planar polynomial solutions also extend directly to explicit, three-dimensional unsteady Navier–Stokes solutions with a symmetry.