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Aumaître S, Fauve S. Intermittency as a Consequence of a Stationarity Constraint on the Energy Flux. PHYSICAL REVIEW LETTERS 2024; 132:114002. [PMID: 38563937 DOI: 10.1103/physrevlett.132.114002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/22/2023] [Accepted: 02/12/2024] [Indexed: 04/04/2024]
Abstract
In his seminal work on turbulence, Kolmogorov made use of the stationary hypothesis to determine the power density spectrum of the velocity field in turbulent flows. However, to our knowledge, the constraints that stationary processes impose on the fluctuations of the energy flux have never been used in the context of turbulence. Here, we recall that the power density spectra of the fluctuations of the injected power, the dissipated power, and the energy flux have to converge to a common value at vanishing frequency. Hence, we show that the intermittent Gledzer-Ohkitani-Yamada (GOY) shell model fulfills these constraints. We argue that they can be related to intermittency. Indeed, we find that the constraint on the fluctuations of the energy flux implies a relation between the scaling exponents that characterize intermittency, which is verified by the GOY shell model and in agreement with the She-Leveque formula. It also fixes the intermittency parameter of the log-normal model at a realistic value. The relevance of these results for real turbulence is drawn in the concluding remarks.
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Affiliation(s)
- Sébastien Aumaître
- SPEC, CEA, CNRS, Université Paris-Saclay, CEA Saclay, 91191, Gif sur Yvette, Cedex, France
| | - Stephan Fauve
- Laboratoire de Physique de l'École Normale Supérieure, CNRS, PSL Research University, Sorbonne Université, Université de Paris, F-75005 Paris, France
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Letournel R, Goudenège L, Zamansky R, Vié A, Massot M. Reexamining the framework for intermittency in Lagrangian stochastic models for turbulent flows: A way to an original and versatile numerical approach. Phys Rev E 2021; 104:015104. [PMID: 34412248 DOI: 10.1103/physreve.104.015104] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 03/23/2021] [Accepted: 05/27/2021] [Indexed: 11/07/2022]
Abstract
The characterization of intermittency in turbulence has its roots in the refined similarity hypotheses of Kolmogorov, and if no proper definition is to be found in the literature, statistical properties of intermittency were studied and models were developed in an attempt to reproduce it. The first contribution of this work is to propose a requirement list to be satisfied by models designed within the Lagrangian framework. Multifractal stochastic processes are a natural choice to retrieve multifractal properties of the dissipation. Among them, we investigate the Gaussian multiplicative chaos formalism, which requires the construction of a log-correlated stochastic process X_{t}. The fractional Gaussian noise of Hurst parameter H=0 is of great interest because it leads to a log correlation for the logarithm of the process. Inspired by the approximation of fractional Brownian motion by an infinite weighted sum of correlated Ornstein-Uhlenbeck processes, our second contribution is to propose a stochastic model: X_{t}=∫_{0}^{∞}Y_{t}^{x}k(x)dx, where Y_{t}^{x} is an Ornstein-Uhlenbeck process with speed of mean reversion x and k is a kernel. A regularization of k(x) is required to ensure stationarity, finite variance, and logarithmic autocorrelation. A variety of regularizations are conceivable, and we show that they lead to the aforementioned multifractal models. To simulate the process, we eventually design a new approach relying on a limited number of modes for approximating the integral through a quadrature X_{t}^{N}=∑_{i=1}^{N}ω_{i}Y_{t}^{x_{i}}, using a conventional quadrature method. This method can retrieve the expected behavior with only one mode per decade, making this strategy versatile and computationally attractive for simulating such processes, while remaining within the proposed framework for a proper description of intermittency.
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Affiliation(s)
- Roxane Letournel
- Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France; Fédération de Mathématiques de CentraleSupélec, CNRS FR-3487, CentraleSupélec, Université Paris-Saclay, 9 rue Joliot Curie, 91190 Gif-sur-Yvette cedex, France; and CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau cedex, France
| | - Ludovic Goudenège
- Fédération de Mathématiques de CentraleSupélec, CNRS FR-3487, CentraleSupélec, Université Paris-Saclay, 9 rue Joliot Curie, 91190 Gif-sur-Yvette cedex, France
| | - Rémi Zamansky
- Institut de Mécanique des Fluides de Toulouse (IMFT), Université de Toulouse, CNRS-INPT-UPS, Toulouse France
| | - Aymeric Vié
- Laboratoire EM2C, CNRS, CentraleSupélec, Université Paris-Saclay, 3 rue Joliot Curie, 91192 Gif-sur-Yvette cedex, France
| | - Marc Massot
- CMAP, CNRS, École Polytechnique, Institut Polytechnique de Paris, Route de Saclay, 91128 Palaiseau cedex, France
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Bos WJT, Laadhari F, Agoua W. Linearly forced isotropic turbulence at low Reynolds numbers. Phys Rev E 2020; 102:033105. [PMID: 33075904 DOI: 10.1103/physreve.102.033105] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 11/28/2019] [Accepted: 08/19/2020] [Indexed: 06/11/2023]
Abstract
We investigate the forcing strength needed to sustain a flow using linear forcing. A critical Reynolds number R_{c} is determined, based on the longest wavelength allowed by the system, the forcing strength and the viscosity. A simple model is proposed for the dissipation rate, leading to a closed expression for the kinetic energy of the flow as a function of the Reynolds number. The dissipation model and the prediction for the kinetic energy are assessed using direct numerical simulations and two-point closure integrations. An analysis of the dissipation-rate equation and the triadic structure of the nonlinear transfer allows to refine the model in order to reproduce the low-Reynolds-number asymptotic behavior, where the kinetic energy is proportional to R-R_{c}.
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Affiliation(s)
- Wouter J T Bos
- LMFA-Ecole Centrale de Lyon, CNRS-Univ. Claude Bernard Lyon 1, Univ. Lyon, 36 Avenue Guy de Collongue, F-69134 Ecully, France
| | - Faouzi Laadhari
- LMFA-Ecole Centrale de Lyon, CNRS-Univ. Claude Bernard Lyon 1, Univ. Lyon, 36 Avenue Guy de Collongue, F-69134 Ecully, France
| | - Wesley Agoua
- LMFA-Ecole Centrale de Lyon, CNRS-Univ. Claude Bernard Lyon 1, Univ. Lyon, 36 Avenue Guy de Collongue, F-69134 Ecully, France
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