More on the asymptotic state of high Reynolds number, smooth-wall turbulent flows

This is an update of a hypothesis (Pullin, Inoue \& Saito, {\it Phys. Fluids}, 2013) concerning the asymptotic state of some canonical, smooth-wall turbulent flows. There it was argued, based on the extrapolation to arbitrarily large Reynolds numbers ($Re_\tau$) of both the log-wake law for the mean velocity profile, and also of Townsend-Perry scaling for stream-wise turbulent velocity fluctuations, that over almost all of the turbulent-flow layer, turbulent velocity fluctuations on outer scales asymptotically decline with increasing $Re_\tau$. Presently this is extended to include vorticity fluctuations using scaling proposed by Panton ({\it Phys. Fluids}, 2009). This suggests that, at least for turbulent channel flow, the asymptotic state consists of vanishingly-small turbulent velocity fluctuations but unbounded enstrophy ($\overline{\omega^2}$) fluctuations on outer scales, over almost the whole turbulent-flow domain.