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Das A, Dutta P, Shukla V. Poles, shocks, and tygers: The time-reversible Burgers equation. Phys Rev E 2024; 109:065108. [PMID: 39020992 DOI: 10.1103/physreve.109.065108] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 10/18/2023] [Accepted: 05/31/2024] [Indexed: 07/20/2024]
Abstract
We construct a formally time-reversible, one-dimensional forced Burgers equation by imposing a global constraint of energy conservation, wherein the constant viscosity is modified to a fluctuating state-dependent dissipation coefficient. The system exhibits dynamical properties which bear strong similarities with those observed for the Burgers equation and can be understood using the dynamics of the poles, shocks, and truncation effects, such as tygers. A complex interplay of these give rise to interesting statistical regimes ranging from hydrodynamic behavior to a completely thermalized warm phase. The end of the hydrodynamic regime is associated with the appearance of a shock in the solution and a continuous transition leading to a truncation-dependent state. Beyond this, the truncation effects such as tygers and the appearance of secondary discontinuity at the resonance point in the solution strongly influence the statistical properties. These disappear at the second transition, at which the global quantities exhibit a jump and attain values that are consistent with the establishment of a quasiequilibrium state characterized by energy equipartition among the Fourier modes. Our comparative analysis shows that the macroscopic statistical properties of the formally time-reversible system and the Burgers equation are equivalent in all the regimes, irrespective of the truncation effects, and this equivalence is not just limited to the hydrodynamic regime, thereby further strengthening the Gallavotti's equivalence conjecture. The properties of the system are further examined by inspecting the complex space singularities in the velocity field of the Burgers equation. Furthermore, an effective theory is proposed to describe the discontinuous transition.
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2
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Maji M, Eswaran KS, Ghosh S, Seshasayanan K, Shukla V. Equivalence of nonequilibrium ensembles: Two-dimensional turbulence with a dual cascade. Phys Rev E 2023; 108:015102. [PMID: 37583143 DOI: 10.1103/physreve.108.015102] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 05/01/2023] [Accepted: 06/16/2023] [Indexed: 08/17/2023]
Abstract
We examine the conjecture of equivalence of nonequilibrium ensembles for turbulent flows in two dimensions in a dual-cascade setup. We construct a formally time-reversible Navier-Stokes equation in two dimensions by imposing global constraints of energy and enstrophy conservation. A comparative study of the statistical properties of its solutions with those obtained from the standard Navier-Stokes equations clearly shows that a formally time-reversible system is able to reproduce the features of a two-dimensional turbulent flow. Statistical quantities based on one- and two-point measurements show an excellent agreement between the two systems for the inverse- and direct-cascade regions. Moreover, we find that the conjecture holds very well for two-dimensional turbulent flows with both conserved energy and enstrophy at finite Reynolds number.
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Affiliation(s)
- Maheswar Maji
- Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, India
| | | | - Sourangshu Ghosh
- Department of Civil Engineering, Indian Institute of Technology Kharagpur, Kharagpur-721 302, India
| | | | - Vishwanath Shukla
- Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur-721 302, India
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Costa G, Barral A, Dubrulle B. Reversible Navier-Stokes equation on logarithmic lattices. Phys Rev E 2023; 107:065106. [PMID: 37464713 DOI: 10.1103/physreve.107.065106] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 01/05/2023] [Accepted: 06/09/2023] [Indexed: 07/20/2023]
Abstract
The three-dimensional reversible Navier-Stokes (RNS) equations are a modification of the dissipative Navier-Stokes (NS) equations, first introduced by Gallavotti [Phys. Lett. A 223, 91 (1996)0375-960110.1016/S0375-9601(96)00729-3], in which the energy or the enstrophy is kept constant by adjusting the viscosity over time. Spectral direct numerical simulations of this model were performed by Shukla et al. [Phys. Rev. E 100, 043104 (2019)2470-004510.1103/PhysRevE.100.043104] and Margazoglou et al. [Phys. Rev. E 105, 065110 (2022)10.1103/PhysRevE.105.065110]. Here we consider a linear, forced reversible system obtained by projecting RNS equations on a log lattice rather than on a linearly spaced grid in Fourier space, as is done in regular spectral numerical simulations. We perform numerical simulations of the system at extremely large resolutions, allowing us to explore regimes of parameters that were out of reach of the direct numerical simulations of Shukla et al. Using the nondimensionalized forcing as a control parameter, and the square root of enstrophy as the order parameter, we confirm the existence of a second-order phase transition well described by a mean-field Landau theory. The log-lattice projection allows us to probe the impact of the resolution, highlighting an imperfect transition at small resolutions with exponents differing from the mean-field predictions. Our findings are in qualitative agreement with predictions of a 1D nonlinear diffusive model, the reversible Leith model of turbulence. We then compare the statistics of the solutions of RNS and NS, in order to shed light on an adaptation of the Gallavotti conjecture, in which there is equivalence of statistics between the reversible and irreversible models, to the case where our reversible model conserves either the enstrophy or the energy. We deduce the conditions in which the two are equivalent. Our results support the validity of the conjecture and represent an instance of nonequilibrium system where ensemble equivalence holds for mean quantities.
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Affiliation(s)
- Guillaume Costa
- Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France
| | - Amaury Barral
- Université Paris-Saclay, CEA, CNRS, SPEC, 91191 Gif-sur-Yvette, France
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Margazoglou G, Biferale L, Cencini M, Gallavotti G, Lucarini V. Nonequilibrium ensembles for the three-dimensional Navier-Stokes equations. Phys Rev E 2022; 105:065110. [PMID: 35854520 DOI: 10.1103/physreve.105.065110] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/27/2021] [Accepted: 05/22/2022] [Indexed: 06/15/2023]
Abstract
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 nonequilibrium 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 hydrodynamical regime, at difference from what was observed previously.
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Affiliation(s)
- G Margazoglou
- Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
- Centre for the Mathematics of Planet Earth, University of Reading, Reading RG6 6AX, United Kingdom
| | - L Biferale
- Department of Physics and INFN, University of Rome Tor Vergata, 00133 Rome, Italy
| | - M Cencini
- Istituto dei Sistemi Complessi, CNR, via dei Taurini 19, I-00185 Rome, Italy
- INFN "Tor Vergata", Via della Ricerca Scientifica 1, 00133 Roma, Italy
| | - G Gallavotti
- INFN, Sezione di Roma and Università "La Sapienza," Piazzale Aldo Moro 2, 00185 Roma, Italy
| | - V Lucarini
- Department of Mathematics and Statistics, University of Reading, Reading RG6 6AX, United Kingdom
- Centre for the Mathematics of Planet Earth, University of Reading, Reading RG6 6AX, United Kingdom
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Jaccod A, Chibbaro S. Constrained Reversible System for Navier-Stokes Turbulence. PHYSICAL REVIEW LETTERS 2021; 127:194501. [PMID: 34797128 DOI: 10.1103/physrevlett.127.194501] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/21/2020] [Revised: 07/13/2021] [Accepted: 09/28/2021] [Indexed: 06/13/2023]
Abstract
Following a Gallavotti's conjecture, stationary states of Navier-Stokes fluids are proposed to be described equivalently by alternative equations besides the Navier-Stokes 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 highly 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 (DNS) at different Reynolds numbers. The possibility of using reversible models to mimic turbulence dynamics is of practical importance for the 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 number. Finally, the theoretical interest in the context of statistical mechanics is briefly discussed.
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Affiliation(s)
- Alice Jaccod
- Sorbonne Université, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
| | - Sergio Chibbaro
- Sorbonne Université, CNRS, UMR 7190, Institut Jean Le Rond d'Alembert, F-75005 Paris, France
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Shukla V, Dubrulle B, Nazarenko S, Krstulovic G, Thalabard S. Phase transition in time-reversible Navier-Stokes equations. Phys Rev E 2019; 100:043104. [PMID: 31770927 DOI: 10.1103/physreve.100.043104] [Citation(s) in RCA: 9] [Impact Index Per Article: 1.8] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2018] [Indexed: 11/07/2022]
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 R_{r}, which quantifies the balance between the fluctuations of kinetic energy at the forcing length scale ℓ_{f} and the total energy E_{0}. For small R_{r}, the RNS equations are found to produce "warm" stationary statistics, e.g., characterized by the partial thermalization of the small scales. For large R_{r}, 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 k_{max}. 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 R_{r} 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 k_{max}. We find that the signatures of the phase transition close to the critical point R_{r}^{★} 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)PYLAAG0375-960110.1016/S0375-9601(96)00729-3] could hold true. We argue that Gallavotti's limit is precisely the joint limit R_{r}→[over >]R_{r}^{★} and h→[over >]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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Affiliation(s)
- Vishwanath Shukla
- Department of Physics, Indian Institute of Technology Kharagpur, Kharagpur 721302, India.,Centre for Theoretical Studies, Indian Institute of Technology Kharagpur, Kharagpur-721302, India.,Université Côte d'Azur, Institut de Physique de Nice (INPHYNI), CNRS UMR 7010, Parc Valrose, 06108 Nice Cedex 2, France
| | - Bérengère Dubrulle
- DSM/IRAMIS/SPEC, CNRS UMR 3680, CEA, Université Paris-Saclay, 91190 Gif sur Yvette, France
| | - Sergey Nazarenko
- Université Côte d'Azur, Institut de Physique de Nice (INPHYNI), CNRS UMR 7010, Parc Valrose, 06108 Nice Cedex 2, France
| | - Giorgio Krstulovic
- Université Côte d'Azur, CNRS, OCA, Laboratoire Lagrange, Bd. de l'Observatoire, 06300 Nice, France
| | - Simon Thalabard
- Instituto Nacional de Matemática Pura e Aplicada, IMPA, 22460-320 Rio de Janeiro, Brazil
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Biferale L, Guido S, Scagliarini A, Toschi F. Topical Issue on Fluids and Structures: Multi-scale coupling and modeling. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2019; 42:28. [PMID: 30848383 DOI: 10.1140/epje/i2019-11808-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Key Words] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/07/2019] [Accepted: 03/07/2019] [Indexed: 06/09/2023]
Affiliation(s)
- Luca Biferale
- Department of Physics and INFN, University of Rome Tor Vergata, Via della Ricerca Scientifica 1, 00133 Rome-I, Rome, Italy
| | - Stefano Guido
- Dipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Università di Napoli "Federico II", P.le V. Tecchio, 80, 80125, Napoli, Italy.
| | - Andrea Scagliarini
- Istituto per le Applicazioni del Calcolo, Consiglio Nazionale delle Ricerche, Via dei Taurini 19, 00185, Roma, Italy
| | - Federico Toschi
- Technische Universiteit Eindhoven, Den Dolech 2, Postbus 513, 5600 MB, Eindhoven, The Netherlands
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Biferale L, Cencini M, De Pietro M, Gallavotti G, Lucarini V. Equivalence of nonequilibrium ensembles in turbulence models. Phys Rev E 2018; 98:012202. [PMID: 30110846 DOI: 10.1103/physreve.98.012202] [Citation(s) in RCA: 10] [Impact Index Per Article: 1.7] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 04/17/2018] [Indexed: 11/07/2022]
Abstract
Understanding under what conditions it is possible to construct equivalent ensembles is key to advancing our ability to connect microscopic and macroscopic properties of nonequilibrium statistical mechanics. In the case of fluid dynamical systems, one issue is to test whether different models for viscosity lead to the same macroscopic properties of the fluid systems in different regimes. Such models include, besides the standard choice of constant viscosity, cases where the time symmetry of the evolution equations is exactly preserved, as it must be in the corresponding microscopic systems, when available. Here a time-reversible dynamics is obtained by imposing the conservation of global observables. We test the equivalence of reversible and irreversible ensembles for the case of a multiscale shell model of turbulence. We verify that the equivalence is obeyed for the mean values of macroscopic observables, up to an error that vanishes as the system becomes more and more chaotic.
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Affiliation(s)
- Luca Biferale
- Dipartimento di Fisica and INFN, Università di Roma "Tor Vergata," Via della Ricerca Scientifica 1, 00133 Roma, Italy
| | - Massimo Cencini
- Istituto dei Sistemi Complessi, CNR, Via dei Taurini 19, 00185 Rome, Italy and INFN "Tor Vergata," Via della Ricerca Scientifica 1, 00133 Roma, Italy
| | - Massimo De Pietro
- Dipartimento di Fisica and INFN, Università di Roma "Tor Vergata," Via della Ricerca Scientifica 1, 00133 Roma, Italy
| | - Giovanni Gallavotti
- INFN, Sezione di Roma and Università "La Sapienza," Piazzale Aldo Moro 2, 00185 Roma, Italy
| | - Valerio Lucarini
- Department of Mathematics and Statistics, University of Reading, Reading RG66AX, United Kingdom.,Centre for the Mathematics of Planet Earth, University of Reading, Reading RG66AX, United Kingdom.,CEN, University of Hamburg, Hamburg 20144, Germany
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