Constrained Reversible System for Navier-Stokes Turbulence

Abstract

Following a Gallavotti's conjecture, stationary states of Navier-Stokes fluids are proposed to be described equivalently by alternative equations besides the NS equation itself. We discuss 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.

Non-equilibrium Ensembles for the three-dimensional Navier-Stokes Equations

“…We remark that, a recently published paper has presented results on the probability of observing negative values of α (see Fig. 4 of [28]) that are in contrast with what has been reported for the hydrodynamical limit (N → ∞ first) in the present work. In particular, in [28] a transition in the shape of the PDF is observed for large numerical grids showing a non negligible probability to have negative events.…”

“…In [28] negative values of α are observed in an R run at N 0 = 1024 and Taylor-based Reynolds number Re λ = 300. Certainly such results are somewhat puzzling and need to be explained/confirmed as we do not expect to be able to observe α < 0 at such large N 0 and such Reynolds number, see subsection B below.…”

Deep Random Vortex Method for Simulation and Inference of Navier-Stokes Equations

“…The ubiquitous Navier-Stokes equations are indispensable for modeling the fluids ranging from meteorology to ocean currents [2,36,39]. Consequently, it is significant to study the Navier-Stokes equation, which has widespread applications in scientific [15], industrial [16], or engineering areas [7]. Recently, with the growth of computer technology and data availability, the development of deep learning techniques has been incredible and shed light on new chances for scientists to press ahead with the research.…”

Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade

“…Studies in three-dimensions (3D) based on reduced models and direct numerical simulations of RNS systems indicate a growing support for the conjecture [5,[8][9][10][11]. A time-reversible shell model of turbulence [8], obtained by imposing a global constraint of energy conservation, was used to explore a part of the RNS parameter space by varying the forcing strength; the system underwent a smooth transition from an equilibrium state to a nonequilibrium stationary state which exhibit an energy cascade from large to small length scales.…”

“…Moreover, the time-reversible shell model was also used to study the time irreversibility that is associated with the nonlinear energy transfers (energy cascade) in 3D, given that the RNS system avoids the explicit time-reversal symmetry breaking by construction [9]. Recently, a model obtained by imposing the constraint that turbulent enstrophy is conserved has been analyzed in 3D turbulence where the RNS and NSE systems were shown to be similar [11].…”