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

## TL;DR

This paper develops explicit polynomial solutions to the unsteady Navier-Stokes equations, serving as benchmarks for numerical methods and analyzing the limitations of common vortex detection criteria in fluid dynamics.

## Contribution

It introduces a new class of exact polynomial solutions for unsteady Navier-Stokes flows and evaluates their effectiveness in flow-feature detection.

## Key findings

- Polynomial solutions reveal deficiencies in streamline-based detection
- Okubo-Weiss criterion's limitations are demonstrated
- Solutions extend to three-dimensional unsteady flows

## Abstract

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.