Phase transition in time-reversible Navier-Stokes equations

Abstract

We present a comprehensive study of the statistical features of a three-dimensional time-reversible Navier-Stokes (RNS) system, wherein the standard viscosity $\nu$ is replaced by a fluctuating thermostat that dynamically compensates for fluctuations in the total energy. We analyze the statistical features of the RNS steady states in terms of a non-negative dimensionless control parameter ${\mathcal{R}}_r$, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale ${\ell }_{\mathrm{f}}$ and the total energy $E_0$. We find that the system exhibits a transition from a high-enstrophy phase at small ${\mathcal{R}}_r$, where truncation effects tend to produce partially thermalized states, to a hydrodynamical phase with low enstrophy at large ${\mathcal{R}}_r$. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous phase transition, where ${\mathcal{R}}_r$ indeed behaves as a thermodynamic control parameter, e.g., a temperature, the enstrophy plays the role of an order parameter, while the symmetry breaking parameter $h$ is (one over) the truncation scale $k_{\max }$. We find that the signatures of the phase transition close to the critical point ${\mathcal{R}}_r^{\star }$ can essentially be deduced from a heuristic mean-field Landau free energy. This point of view allows us to reinterpret the relevant asymptotics in which the dynamical ensemble equivalence conjectured by Gallavotti, Phys.Lett.A, 223, 1996 could hold true. Our numerics indicate that the low-order statistics of the 3D RNS are indeed qualitatively similar to those observed in direct numerical simulations of the standard Navier-Stokes (NS) equations with viscosity chosen so as to match the average value of the reversible viscosity.

arXiv commentRevised version; results unchanged

References in corpus (22)

• Dynamical Ensembles in Nonequilibrium Statistical Mechanics
• Dynamical ensembles in stationary states
• Macroscopic fluctuation theory for stationary non equilibrium states
• Hyperviscosity, Galerkin truncation and bottlenecks in turbulence
• Effective Dissipation and Turbulence in Spectrally Truncated Euler Flows
• Equivalence and nonequivalence of ensembles: Thermodynamic, macrostate, and measure levels
• Non-equilibrium work relations
• Cascades, thermalization and eddy viscosity in helical Galerkin truncated Euler flows
• Time-symmetry breaking in turbulence
• Dynamical ensembles equivalence in fluid mechanics
• Equivalence of dynamical ensembles and Navier Stokes equations
• Fluctuations in 2D reversibly-damped turbulence
• Steep cliffs and saturated exponents in three dimensional scalar turbulence
• Deriving GENERIC from a generalized fluctuation symmetry
• An optimization principle for deriving nonequilibrium statistical models of Hamiltonian dynamics
• Time-reversible Dynamical Systems for Turbulence
• Two-fluid model of the truncated Euler Equations
• Equivalence of nonequilibrium ensembles in turbulence models
• Heavy inertial particles in turbulent flows gain energy slowly but lose it rapidly
• Time irreversibility in reversible shell models of turbulence
• Optimal response to non-equilibrium disturbances under truncated Burgers-Hopf dynamics
• Mixed flux-equipartition solutions of a diffusion model of nonlinear cascades

Cited by in corpus (11)

• Variable energy flux in turbulence
• Turbulent cascade, bottleneck and thermalized spectrum in hyperviscous flows
• Constrained Reversible system for Navier-Stokes Turbulence: evidence for Gallavotti's equivalence conjecture
• Statistics of Lagrangian Trajectories in a Rotating Turbulent Flow
• Equivalent Ensembles, Turbulence and Fluctuation Theorem
• A minimal phase-coupling model for intermittency in turbulent systems
• Viscosity, Reversibillity, Chaotic Hypothesis, Fluctuation Theorem and Lyapunov Pairing
• Non-equilibrium Ensembles for the three-dimensional Navier-Stokes Equations
• Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade
• Poles, Shocks and Tygers: The Time-reversible Burgers equation
• Behind the mirror: the hidden dissipative singular solutions of ideal reversible fluids on log-lattices