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Fuchs A, Herbert C, Rolland J, Wächter M, Bouchet F, Peinke J. Instantons and the Path to Intermittency in Turbulent Flows. PHYSICAL REVIEW LETTERS 2022; 129:034502. [PMID: 35905359 DOI: 10.1103/physrevlett.129.034502] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/14/2021] [Revised: 02/23/2022] [Accepted: 06/10/2022] [Indexed: 06/15/2023]
Abstract
Processes leading to anomalous fluctuations in turbulent flows, referred to as intermittency, are still challenging. We consider cascade trajectories through scales as realizations of a stochastic Langevin process for which multiplicative noise is an intrinsic feature of the turbulent state. The trajectories are conditioned on their entropy exchange. Such selected trajectories concentrate around an optimal path, called instanton, which is the minimum of an effective action. The action is derived from the Langevin equation, estimated from measured data. In particular instantons with negative entropy pinpoint the trajectories responsible for the emergence of non-Gaussian statistics at small scales.
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Affiliation(s)
- A Fuchs
- Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany
| | - C Herbert
- Ens de Lyon, CNRS, Laboratoire de Physique, F-69364 Lyon, France
| | - J Rolland
- Université de Lille, CNRS, ONERA, Arts et Métiers Institute of Technology, Centrale Lille, UMR 9014 - LMFL - Laboratoire de Mécanique des fluides de Lille - Kampé de Fériet, F-59000 Lille, France
| | - M Wächter
- Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany
| | - F Bouchet
- Ens de Lyon, CNRS, Laboratoire de Physique, F-69364 Lyon, France
| | - J Peinke
- Institute of Physics and ForWind, University of Oldenburg, Küpkersweg 70, 26129 Oldenburg, Germany
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Corbetta A, Menkovski V, Benzi R, Toschi F. Deep learning velocity signals allow quantifying turbulence intensity. SCIENCE ADVANCES 2021; 7:7/12/eaba7281. [PMID: 33731341 PMCID: PMC7968843 DOI: 10.1126/sciadv.aba7281] [Citation(s) in RCA: 3] [Impact Index Per Article: 1.0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Figures] [Subscribe] [Scholar Register] [Received: 01/08/2020] [Accepted: 01/29/2021] [Indexed: 06/12/2023]
Abstract
Turbulence, the ubiquitous and chaotic state of fluid motions, is characterized by strong and statistically nontrivial fluctuations of the velocity field, and it can be quantitatively described only in terms of statistical averages. Strong nonstationarities impede statistical convergence, precluding quantifying turbulence, for example, in terms of turbulence intensity or Reynolds number. Here, we show that by using deep neural networks, we can accurately estimate the Reynolds number within 15% accuracy, from a statistical sample as small as two large-scale eddy turnover times. In contrast, physics-based statistical estimators are limited by the convergence rate of the central limit theorem and provide, for the same statistical sample, at least a hundredfold larger error. Our findings open up previously unexplored perspectives and the possibility to quantitatively define and, therefore, study highly nonstationary turbulent flows as ordinarily found in nature and in industrial processes.
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Affiliation(s)
- Alessandro Corbetta
- Department of Applied Physics, Eindhoven University of Technology, Eindhoven, Netherlands
| | - Vlado Menkovski
- Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands
| | - Roberto Benzi
- Department of Physics, University of Rome Tor Vergata, Rome, Italy
| | - Federico Toschi
- Department of Applied Physics, Eindhoven University of Technology, Eindhoven, Netherlands.
- Department of Mathematics and Computer Science, Eindhoven University of Technology, Eindhoven, Netherlands
- CNR-IAC, Rome, Italy
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3
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Margazoglou G, Biferale L, Grauer R, Jansen K, Mesterházy D, Rosenow T, Tripiccione R. Hybrid Monte Carlo algorithm for sampling rare events in space-time histories of stochastic fields. Phys Rev E 2019; 99:053303. [PMID: 31212557 DOI: 10.1103/physreve.99.053303] [Citation(s) in RCA: 3] [Impact Index Per Article: 0.6] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/05/2018] [Indexed: 11/07/2022]
Abstract
We introduce a variant of the Hybrid Monte Carlo (HMC) algorithm to address large-deviation statistics in stochastic hydrodynamics. Based on the path-integral approach to stochastic (partial) differential equations, our HMC algorithm samples space-time histories of the dynamical degrees of freedom under the influence of random noise. First, we validate and benchmark the HMC algorithm by reproducing multiscale properties of the one-dimensional Burgers equation driven by Gaussian and white-in-time noise. Second, we show how to implement an importance sampling protocol to significantly enhance, by orders of magnitudes, the probability to sample extreme and rare events, making it possible to estimate moments of field variables of extremely high order (up to 30 and more). By employing reweighting techniques, we map the biased configurations back to the original probability measure in order to probe their statistical importance. Finally, we show that by biasing the system towards very intense negative gradients, the HMC algorithm is able to explore the statistical fluctuations around instanton configurations. Our results will also be interesting and relevant in lattice gauge theory since they provide unique insights into reweighting techniques.
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Affiliation(s)
- G Margazoglou
- Department of Physics, University of Rome Tor Vergata and INFN-Tor Vergata, 00133 Rome, Italy.,Computation-based Science and Technology Research Center, Cyprus Institute, 2121 Nicosia, Cyprus
| | - L Biferale
- Department of Physics, University of Rome Tor Vergata and INFN-Tor Vergata, 00133 Rome, Italy
| | - R Grauer
- Institut für Theoretische Physik I, Ruhr-University Bochum, 44780 Bochum, Germany
| | - K Jansen
- NIC, DESY, 15738 Zeuthen, Germany
| | - D Mesterházy
- Institute for Theoretical Physics, University of Bern, 3012 Bern, Switzerland
| | - T Rosenow
- Institut für Physik, Brandenburg University of Technology Cottbus-Senftenberg, 03046 Cottbus, Germany
| | - R Tripiccione
- Department of Physics, Università di Ferrara and INFN-Ferrara, 44122 Ferrara, Italy
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Ebener L, Margazoglou G, Friedrich J, Biferale L, Grauer R. Instanton based importance sampling for rare events in stochastic PDEs. CHAOS (WOODBURY, N.Y.) 2019; 29:063102. [PMID: 31266309 DOI: 10.1063/1.5085119] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.4] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/09/2018] [Accepted: 05/10/2019] [Indexed: 06/09/2023]
Abstract
We present a new method for sampling rare and large fluctuations in a nonequilibrium system governed by a stochastic partial differential equation (SPDE) with additive forcing. To this end, we deploy the so-called instanton formalism that corresponds to a saddle-point approximation of the action in the path integral formulation of the underlying SPDE. The crucial step in our approach is the formulation of an alternative SPDE that incorporates knowledge of the instanton solution such that we are able to constrain the dynamical evolutions around extreme flow configurations only. Finally, a reweighting procedure based on the Girsanov theorem is applied to recover the full distribution function of the original system. The entire procedure is demonstrated on the example of the one-dimensional Burgers equation. Furthermore, we compare our method to conventional direct numerical simulations as well as to Hybrid Monte Carlo methods. It will be shown that the instanton-based sampling method outperforms both approaches and allows for an accurate quantification of the whole probability density function of velocity gradients from the core to the very far tails.
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Affiliation(s)
- Lasse Ebener
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - Georgios Margazoglou
- Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, I-00133 Roma, Italy
| | - Jan Friedrich
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
| | - Luca Biferale
- Department of Physics and INFN, University of Rome "Tor Vergata," Via della Ricerca Scientifica 1, I-00133 Roma, Italy
| | - Rainer Grauer
- Institut für Theoretische Physik I, Ruhr-Universität Bochum, Universitätsstraße 150, 44780 Bochum, Germany
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Gürcan ÖD. Nested polyhedra model of isotropic magnetohydrodynamic turbulence. Phys Rev E 2018; 97:063111. [PMID: 30011494 DOI: 10.1103/physreve.97.063111] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 12/22/2017] [Indexed: 06/08/2023]
Abstract
A nested polyhedra model has been developed for magnetohydrodynamic turbulence. Driving only the velocity field at large scales with random, divergence-free forcing results in a clear, stationary k^{-5/3} spectrum for both kinetic and magnetic energies. Since the model naturally effaces disparate scale interactions, does not have a guide field, and avoids injecting any sign of helicity by random forcing, the resulting three-dimensional k spectrum is statistically isotropic. The strengths and weaknesses of the model are demonstrated by considering large or small magnetic Prandtl numbers. It was also observed that the timescale for the equipartition offset with those of the smallest scales shows a k^{-1/2} scaling.
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Affiliation(s)
- Ö D Gürcan
- CNRS, Laboratoire de Physique des Plasmas, Ecole Polytechnique, 91128 Palaiseau, France and Sorbonne Universités, UPMC Univ Paris 06, Paris, France
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De Pietro M, Biferale L, Mailybaev AA. Inverse energy cascade in nonlocal helical shell models of turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2015; 92:043021. [PMID: 26565346 DOI: 10.1103/physreve.92.043021] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 08/27/2015] [Indexed: 06/05/2023]
Abstract
Following the exact decomposition in eigenstates of helicity for the Navier-Stokes equations in Fourier space [F. Waleffe, Phys. Fluids A 4, 350 (1992)], we introduce a modified version of helical shell models for turbulence with nonlocal triadic interactions. By using both an analytical argument and numerical simulation, we show that there exists a class of models, with a specific helical structure, that exhibits a statistically stable inverse energy cascade, in close analogy with that predicted for the Navier-Stokes equations restricted to the same helical interactions. We further support the idea that turbulent energy transfer is the result of a strong entanglement among triads possessing different transfer properties.
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Affiliation(s)
- Massimo De Pietro
- Dip. di Fisica and INFN, Università "Tor Vergata," Via della Ricerca Scientifica 1, I-00133 Roma, Italy
| | - Luca Biferale
- Dip. di Fisica and INFN, Università "Tor Vergata," Via della Ricerca Scientifica 1, I-00133 Roma, Italy
| | - Alexei A Mailybaev
- Instituto Nacional de Matemática Pura e Aplicada-IMPA, Est. Dona Castorina 110, Rio de Janeiro 22460-320 Brazil
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Mailybaev AA. Blowup as a driving mechanism of turbulence in shell models. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2013; 87:053011. [PMID: 23767624 DOI: 10.1103/physreve.87.053011] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.2] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 03/02/2013] [Indexed: 06/02/2023]
Abstract
Since Kolmogorov proposed his phenomenological theory of hydrodynamic turbulence in 1941, the description of the mechanism leading to the energy cascade and anomalous scaling remains an open problem in fluid mechanics. Soon after, in 1949, Onsager noticed that the scaling properties in the inertial range imply nondifferentiability of the velocity field in the limit of vanishing viscosity. This observation suggests that the turbulence mechanism may be related to a finite-time singularity (blowup) of incompressible Euler equations. However, the existence of such blowup is still an open problem too. In this paper, we show that the blowup indeed represents the driving mechanism of the inertial range for a simplified (shell) model of turbulence. Here, blowups generate coherent structures (instantons), which travel through the inertial range in finite time and are described by universal self-similar statistics. The anomaly (deviation of scaling exponents of velocity moments from the Kolmogorov theory) is related analytically to the process of instanton creation using the large deviation principle. The results are confirmed by numerical simulations.
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Affiliation(s)
- Alexei A Mailybaev
- Instituto Nacional de Matemática Pura e Aplicada-IMPA, Rio de Janeiro, Brazil.
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Mailybaev AA. Computation of anomalous scaling exponents of turbulence from self-similar instanton dynamics. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 86:025301. [PMID: 23005816 DOI: 10.1103/physreve.86.025301] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 06/09/2012] [Indexed: 06/01/2023]
Abstract
We show that multiscaling properties of developed turbulence in shell models, which lead to anomalous scaling exponents in the inertial range, are determined exclusively by instanton dynamics. Instantons represent correlated extreme events localized in space-time, whose structure is described by self-similar statistics with a single universal scaling exponent. We show that anomalous scaling exponents appear due to the process of instanton creation. A simplified model of instanton creation is suggested, which adequately describes this anomaly.
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Affiliation(s)
- Alexei A Mailybaev
- Instituto Nacional de Matemática Pura e Aplicada-IMPA, Rio de Janeiro, Brazil.
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Mailybaev AA. Renormalization and universality of blowup in hydrodynamic flows. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2012; 85:066317. [PMID: 23005216 DOI: 10.1103/physreve.85.066317] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [MESH Headings] [Track Full Text] [Subscribe] [Scholar Register] [Received: 01/10/2012] [Revised: 03/16/2012] [Indexed: 06/01/2023]
Abstract
We consider self-similar solutions describing intermittent bursts in shell models of turbulence and study their relationship with blowup phenomena in continuous hydrodynamic models. First, we show that these solutions are very close to self-similar solution for the Fourier transformed inviscid Burgers equation corresponding to shock formation from smooth initial data. Then, the result is generalized to hyperbolic conservation laws in one space dimension describing compressible flows. It is shown that the renormalized wave profile tends to a universal function, which is independent both of initial conditions and of a specific form of the conservation law. This phenomenon can be viewed as a new manifestation of the renormalization group theory. Finally, we discuss possibilities for application of the developed theory for detecting and describing a blowup in incompressible flows.
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Affiliation(s)
- Alexei A Mailybaev
- Instituto Nacional de Matemática Pura e Aplicada-IMPA, Rio de Janeiro, Brazil.
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Daumont I, Peyrard M. One-dimensional "turbulence" in a discrete lattice. CHAOS (WOODBURY, N.Y.) 2003; 13:624-636. [PMID: 12777127 DOI: 10.1063/1.1530991] [Citation(s) in RCA: 2] [Impact Index Per Article: 0.1] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/24/2023]
Abstract
We study a one-dimensional discrete analog of the von Karman flow, widely investigated in turbulence. A lattice of anharmonic oscillators is excited by both ends in order to create a large scale structure in a highly nonlinear medium, in the presence of a dissipative term proportional to the second order finite difference of the velocities, similar to the viscous term in a fluid. In a first part, the energy density is investigated in real and Fourier space in order to characterize the behavior of the system on a local scale. At low amplitude of excitation the large scale structure persists in the system but all modes are however excited and exchange energy, leading to a power law spectrum for the energy density, which is remarkably stable against changes in the model parameters, amplitude of excitation, or damping. In the spirit of shell models, this regime can be described in terms of interacting scales. At higher amplitude of excitation, the large scale structure is destroyed and the dynamics of the system can be viewed as resulting from the creation, interaction, and decay of localized excitations, the discrete breathers, the one-dimensional equivalents of vortices in a fluid. The spectrum of the energy density is well described by the spectrum of the breathers, and shows an exponential decay with the wave vector. Due to this exponential behavior, the spectrum is dominated by the most intense breathers. In this regime, the probability distribution of the increments of velocity between neighboring points is remarkably similar to the experimental results of turbulence and can be described by distributions deduced from nonextensive thermodynamics as in fluids. In a second part the power dissipated in the whole lattice is studied to characterize the global behavior of the system. Its probability distribution function shows non-Gaussian fluctuations similar to the one exhibited recently in a large class of "inertial systems," i.e., systems that cannot be divided into mesoscopic regions which are independent. The properties of the nonlinear excitations of the lattice provide a partial understanding of this behavior.
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Affiliation(s)
- Isabelle Daumont
- Laboratoire de Physique, Ecole Normale Superieure de Lyon, 46 allie d'Italie, 69364 Lyon Cedex 07, France
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L'vov VS. Quasisolitons and asymptotic multiscaling in shell models of turbulence. PHYSICAL REVIEW. E, STATISTICAL, NONLINEAR, AND SOFT MATTER PHYSICS 2002; 65:026309. [PMID: 11863655 DOI: 10.1103/physreve.65.026309] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/08/2001] [Indexed: 05/23/2023]
Abstract
A variation principle is suggested to find self-similar solitary solutions (referred to as solitons) of shell model of turbulence. For the Sabra shell model the shape of the solitons is approximated by rational trial functions with relative accuracy of O(10(-3)). It is found how the soliton shape, propagation time t(n) (from a shell n to shells with n --> infinity), and the dynamical exponent z(0) (which governs the time rescaling of the solitons in different shells) depend on parameters of the model. For a finite interval of z the author discovered quasisolitons which approximate with high accuracy corresponding self-similar equations for an interval of times from -infinity to some time in the vicinity of the peak maximum or even after it. The conjecture is that the trajectories in the vicinity of the quasisolitons (with continuous spectra of z) provide an essential contribution to the multiscaling statistics of high-order correlation functions, referred to in the paper as an asymptotic multiscaling. This contribution may be even more important than that of the trajectories in the vicinity of the exact soliton with a fixed value z(0). Moreover there are no solitons in some regions of the parameters where quasisolitons provide a dominant contribution to the asymptotic multiscaling.
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Affiliation(s)
- Victor S L'vov
- Department of Chemical Physics, The Weizmann Institute of Science, Rehovot 76100, Israel
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