Spatio-temporal correlations in three-dimensional homogeneous and isotropic turbulence

Gorbunova, Anastasiia; Balarac, Guillaume; Canet, Léonie; Eyink, Gregory L.; Rossetto, Vincent

doi:10.1063/5.0046677

Cited by 13 publications

(21 citation statements)

References 44 publications

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“…At the origin of the system, the correlation time of the solution is larger, but it rapidly decays as one moves away from this point. One can contrast this picture with turbulent velocity fields in the Eulerian setting, in which the same local roughness in time and in space is observed [30,31].…”

Section: Discussionmentioning

confidence: 99%

“…In turbulent velocity fields, the Eulerian structure function in time (that is, increments in time measured at some fixed point in space) show the same local regularity as their spatial equivalent, with a power-law structure function approximately given by (for the second order) E(δ τ u) 2 ∝ τ 2/3 in the limit of vanishing viscosity, for a small time interval τ . This effect, which has been measured in direct numerical simulations [30,31], can be phenomenologically explained with the sweeping effect [32], the random advection of the small scales by the large scale motion of the flow, and reproduced with spatio-temporal random fields [33,34].…”

Section: Temporal Statisticsmentioning

confidence: 99%

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Space-time statistics of a linear dynamical energy cascade model

Apolinário,

Chevillard

2021

Preprint

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A linear dynamical model for the development of the turbulent energy cascade was introduced in Apolinário et al [arxiv 2107.03309]. This partial differential equation, randomly stirred by a forcing term which is smooth in space and deltacorrelated in time, was shown to converge at infinite time towards a state of finite variance, without the aid of viscosity. Furthermore, the spatial profile of its solution gets rough, with the same regularity as a fractional Gaussian field. We here focus on the temporal behavior and derive explicit asymptotic predictions for the correlation function in time of this solution and observe that their regularity is not influenced by the spatial regularity of the problem, only by the correlation in time of the stirring contribution. We also show that the correlation in time of the solution depends on the position, contrary to its correlation in space at fixed times. We then investigate the influence of a forcing which is correlated in time on the spatial and time statistics of this equation. In this situation, while for small correlation times the homogeneous spatial statistics of the white-in-time case are recovered, for large correlation times homogeneity is broken, and a concentration around the origin of the system is observed in the velocity profiles. In other words, this fractional velocity field is a representation in one-dimension, through a linear dynamical model, of the self-similar velocity fields proposed by Kolmogorov in 1941, but only at fixed times, for a delta-correlated forcing, in which case the spatial statistics is homogeneous and rough, as expected of a turbulent velocity field. The regularity in time of turbulence, however, is not captured by this model.

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Section: Discussionmentioning

confidence: 99%

Section: Temporal Statisticsmentioning

confidence: 99%

Space-time statistics of a linear dynamical energy cascade model

Apolinário,

Chevillard

2021

Preprint

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“…Kraichnan's model assumes stationary turbulence and sets the sweeping velocity to the rms velocity divided by √ 3, under the hypothesis of statistical isotropy [40,73]. Since this model has good numerical support in the inertial range [45,70,71], any good model should reduce to…”

Section: Kraichnan Random Sweeping Approximationmentioning

confidence: 99%

Generation of gravitational waves from freely decaying turbulence

Auclair,

Caprini,

Cutting

et al. 2022

Preprint

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We study the stochastic gravitational wave background (SGWB) produced by freely decaying vortical turbulence in the early Universe. We thoroughly investigate the time correlation of the velocity field, and hence of the anisotropic stresses producing the gravitational waves. By direct numerical simulation, we show that the unequal time correlation function (UETC) of the Fourier components of the velocity field is Gaussian in the time difference, as predicted by the "sweeping" decorrelation model. We introduce a decorrelation model that can be extended to wavelengths around the integral scale of the flow. Supplemented with the evolution laws of the kinetic energy and of the integral scale, this provides a new model UETC of the turbulent velocity field consistent with the simulations. We discuss the UETC as a positive definite kernel, and propose to use the Gibbs kernel for the velocity UETC as a natural way to ensure positive definiteness of the SGWB. The SGWB is given by a 4-dimensional integration of the resulting anisotropic stress UETC with the gravitational wave Green's function. We perform this integration using a Monte Carlo algorithm based on importance sampling, and find that the result matches that of the direct numerical simulations. Furthermore, the SGWB obtained from the numerical integration and from the simulations show close agreement with a model in which the source is constant in time and abruptly turns off after a few eddy turnover times. Based on this assumption, we provide an approximate analytical form for the SGWB spectrum and its scaling with the initial kinetic energy and integral scale. Finally, we use our model and numerical integration algorithm to show that including an initial growth phase for the turbulent flow heavily influences the spectral shape of the SGWB. This highlights the importance of a complete understanding of the turbulence generation mechanism.

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“…This large-time regime had not been predicted before. However, it was already in germ in early studies of sweeping, as was noted in Gorbunova et al (2021a). This allows one to provide a simple physical interpretation of these two regimes, see § 7.4.…”

Section: Large-time Regimementioning

confidence: 99%

“…Let us now provide an heuristic derivation of these results, proposed in Gorbunova et al (2021a). The early analysis of Eulerian sweeping effects by (Kraichnan 1964) was based on the Lagrangian expression for the Eulerian velocity field as…”

Section: Intuitive Physical Interpretationmentioning

confidence: 99%

Functional renormalisation group for turbulence

Canet

2022

J. Fluid Mech.

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Turbulence is a complex nonlinear and multi-scale phenomenon. Although the fundamental underlying equations, the Navier–Stokes equations, have been known for two centuries, it remains extremely challenging to extract from them the statistical properties of turbulence. Therefore, for practical purposes, a sustained effort has been devoted to obtaining an effective description of turbulence, that we may call turbulence modelling, or statistical theory of turbulence. In this respect, the renormalisation group (RG) appears as a tool of choice, since it is precisely designed to provide effective theories from fundamental equations by performing in a systematic way the average over fluctuations. However, for Navier–Stokes turbulence, a suitable framework for the RG, allowing in particular for non-perturbative approximations, has been missing, which has thwarted RG applications for a long time. This framework is provided by the modern formulation of the RG called the functional renormalisation group (FRG). The use of the FRG has enabled important progress in the theoretical understanding of homogeneous and isotropic turbulence. The major one is the rigorous derivation, from the Navier–Stokes equations, of an analytical expression for any Eulerian multi-point multi-time correlation function, which is exact in the limit of large wavenumbers. We propose in this JFM Perspectives article a survey of the FRG method for turbulence. We provide a basic introduction to the FRG and emphasise how the field-theoretical framework allows one to systematically and profoundly exploit the symmetries. We stress that the FRG enables one to describe fully developed turbulence forced at large scales, which was not accessible by perturbative means. We show that it yields the energy spectrum and second-order structure function with accurate estimates of the related constants, and also the behaviour of the spectrum in the near-dissipative range. Finally, we expound the derivation of the spatio-temporal behaviour of $n$ -point correlation functions, and largely illustrate these results through the analysis of data from experiments and direct numerical simulations.

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Spatio-temporal correlations in three-dimensional homogeneous and isotropic turbulence

Cited by 13 publications

References 44 publications

Space-time statistics of a linear dynamical energy cascade model

Space-time statistics of a linear dynamical energy cascade model

Generation of gravitational waves from freely decaying turbulence

Functional renormalisation group for turbulence

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