  Solutions of the forced Navier-Stokes equation have been conjectured to
thermalize at scales larger than the forcing scale, similar to an absolute
equilibrium obtained for the spectrally-truncated Euler equation. Using direct
numeric simulations of Taylor-Green flows and general-periodic helical flows,
we present results on the probability density function, energy spectrum,
auto-correlation function and correlation time that compare the two systems. In
the case of highly helical flows, we derive an analytic expression describing
the correlation time for the absolute equilibrium of helical flows that is
different from the $E^{-1/2}k^{-1}$-scaling law of weakly helical flows. This
model predicts a new helicity-based scaling law for the correlation time as
$\tau(k)\sim H^{-1/2}k^{-1/2}$. This scaling law is verified in simulations of
the truncated Euler equation. In simulations of the Navier-Stokes equations the
large scale modes of forced Taylor-Green symmetric flows (with zero total
helicity and large separation of scales) follow the same properties as absolute
equilibrium including a $\tau(k) \sim E^{-1/2}k^{-1}$ scaling for the
correlation time. General-periodic helical flows also show similarities between
the two systems, however the largest scales of the forced flows deviate from
the absolute equilibrium solutions.
