Constrained Reversible system for Navier-Stokes Turbulence: evidence for Gallavotti's equivalence conjecture

Abstract

Following the Gallavotti's conjecture, Stationary states of Navier-Stokes fluids are proposed to be described equivalently by alternative equations besides the NS equation itself. We propose a model system symmetric under time-reversal based on the Navier-Stokes equations constrained to keep the Enstrophy constant. It is demonstrated through high-resolved numerical experiments that the reversible model evolves to a stationary state which reproduces quite accurately all statistical observables relevant for the physics of turbulence extracted by direct numerical simulations at different Reynolds numbers. The possibility of using reversible models to mimic turbulence dynamics is of practical importance for coarse-grained version of Navier-Stokes equations, as used in Large-eddy simulations. Furthermore, the reversible model appears mathematically simpler, since enstrophy is bounded to be constant for every Reynolds. Finally, the theoretically interest in the context of statistical mechanics is briefly discussed.

Viscosity, Reversibillity, Chaotic Hypothesis, Fluctuation Theorem and Lyapunov Pairing

“…So the same argument can simply be extended to many other equations in which the size of a parameter controls the increasingly "chaotic" motion of a system. Examples of this phenomenon have been explicitly considered adding new examples to a wide literature of homogenization phenomena: see [16,17,24,26] for fluid equations or [5,17,22]. Thus the conjecture in Sec.IV although quite unsatisfactory, as pointed out, seems to hold in its generality, [22,24].…”