  At the molecular level fluid motions are, by first principles, described by
time reversible laws. On the other hand, the coarse grained macroscopic
evolution is suitably described by the Navier-Stokes equations, which are
inherently irreversible, due to the dissipation term. Here, a reversible
version of three-dimensional Navier-Stokes is studied, by introducing a
fluctuating viscosity constructed in such a way that enstrophy is conserved,
along the lines of the paradigm of microcanonical versus canonical treatment in
equilibrium statistical mechanics. Through systematic simulations we attack two
important questions: (a) What are the conditions that must be satisfied in
order to have a statistical equivalence between the two non-equilibrium
ensembles? (b) What is the empirical distribution of the fluctuating viscosity
observed by changing the Reynolds number and the number of modes used in the
discretization of the evolution equation? The latter point is important also to
establish regularity conditions for the reversible equations. We find that the
probability to observe negative values of the fluctuating viscosity becomes
very quickly extremely small when increasing the effective Reynolds number of
the flow in the fully resolved hydro dynamical regime, at difference from what
was observed previously.
