  We report results on the behavior of a particular incompressible
Navier-Stokes (NS) flow in the whole space $\R^{3}$, related to the complex
singular solutions introduced by Li and Sinai in \cite{LiSi08} that blow up at
a finite time. The flow exhibits for some time a tornado-like behavior: a sharp
increase of vorticity and of the maximal velocity, with concentration in an
annular region around an axis. There is however no rotation around the axis and
the approximate axial symmetry with no swirl excludes a real blow-up. We
conclude with a discussion on the possible candidates for a real blow-up.
