Phase transition in time-reversible Navier-Stokes equations

Vishwanath Shukla, Bérengère Dubrulle, Sergey Nazarenko, Giorgio Krstulovic, and Simon Thalabard
Phys. Rev. E 100, 043104 – Published 9 October 2019

Abstract

We present a comprehensive study of the statistical features of a three-dimensional (3D) time-reversible truncated Navier-Stokes (RNS) system, wherein the standard viscosity ν 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 Rr, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale f and the total energy E0. For small Rr, the RNS equations are found to produce “warm” stationary statistics, e.g., characterized by the partial thermalization of the small scales. For large Rr, the stationary solutions have features akin to standard hydrodynamic ones: they have compact energy support in k space and are essentially insensitive to the truncation scale kmax. The transition between the two statistical regimes is observed to be smooth but rather sharp. Using insights from a diffusion model of turbulence (Leith model), we argue that the transition is in fact akin to a continuous second-order phase transition, where Rr indeed behaves as a thermodynamic control parameter, e.g., a temperature. A relevant order parameter can be suitably defined in terms of a (normalized) enstrophy, while the symmetry-breaking parameter h is identified as (one over) the truncation scale kmax. We find that the signatures of the phase transition close to the critical point Rr 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, 91 (1996)] could hold true. We argue that Gallavotti's limit is precisely the joint limit Rr>Rr and h>0, with the overset symbol “>” indicating that those limits are approached from above. The limit therefore relates to the statistical features at the critical point. In this regime, 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 equations with viscosity chosen so as to match the average value of the reversible thermostat. This result suggests that Gallavotti's equivalence conjecture could indeed be of relevance to model 3D turbulent statistics, and provides a clear guideline for further numerical investigations involving higher resolutions.

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  • Received 28 November 2018
  • Revised 20 July 2019

DOI:https://doi.org/10.1103/PhysRevE.100.043104

©2019 American Physical Society

Physics Subject Headings (PhySH)

Fluid DynamicsStatistical Physics & ThermodynamicsNonlinear Dynamics

Authors & Affiliations

Vishwanath Shukla1,2,3,*, Bérengère Dubrulle4, Sergey Nazarenko3, Giorgio Krstulovic5, and Simon Thalabard6

  • 1Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India
  • 2Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur-721302, India
  • 3Université Côte d'Azur, Institut de Physique de Nice (INPHYNI), CNRS UMR 7010, Parc Valrose, 06108 Nice Cedex 2, France
  • 4DSM/IRAMIS/SPEC, CNRS UMR 3680, CEA, Université Paris-Saclay, 91190 Gif sur Yvette, France
  • 5Université Côte d'Azur, CNRS, OCA, Laboratoire Lagrange, Bd. de l'Observatoire, 06300 Nice, France
  • 6Instituto Nacional de Matemática Pura e Aplicada, IMPA, 22460-320 Rio de Janeiro, Brazil

  • *research.vishwanath@gmail.com

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Issue

Vol. 100, Iss. 4 — October 2019

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