Elementary properties of the enstrophy and strain fields in confined two-dimensional flows

The evolution of decaying two-dimensional turbulent flows in a bounded domain is considered. It is shown that the global enstrophy is always equal to the total squared strain field, when integrated over a domain with no-slip boundaries, despite the complex evolution of the flow and strong vortex-wall interactions. This property is also valid for a square domain with stress-free walls, and in general for any polygonal boundary. In contrast, the enstrophy is always greater than the squared strain field in a stress-free circular domain, and in general for any closed domain with negatively-signed curvature at all points of the boundary.