Is turbulence ergodic?

Galanti, Barak; Tsinober, A.

doi:10.1016/j.physleta.2004.07.009

Cited by 42 publications

(31 citation statements)

References 6 publications

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“…This is the so-called dissipation range in which most kinetic energy is lost due to viscous friction. The Kolmogorov spectrum is confirmed by numerous measurements and simulations [18][19][20][21]. For turbulent flows, the energy flux Π(κ) is also defined which shows the rate of energy transfer at the different wavenumbers κ.…”

Section: Introductionmentioning

confidence: 60%

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

Kalmár‐Nagy

Bak

2019

Nonlinear Dyn

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In Richardson's cascade description of turbulence, large vortices break up to form smaller ones, transferring the kinetic energy of the flow from large to small scales. This energy is dissipated at the smallest scales due to viscosity. We study energy cascade in a phenomenological model of vortex breakdown. The model is a binary tree of spring-connected masses, with dampers acting on the lowest level. The masses and stiffnesses between levels change according to a power law. The different levels represent different scales, enabling the definition of "mass wavenumbers." The eigenvalue distribution of the model exhibits a devil's staircase self-similarity. The energy spectrum of the model (defined as the energy distribution among the different mass wavenumbers) is derived in the asymptotic limit. A decimation procedure is applied to replace the model with an equivalent chain oscillator. For a significant range of the stiffness decay parameter, the energy spectrum is qualitatively similar to the Kolmogorov spectrum of 3D homogeneous, isotropic turbulence and we find the stiffness parameter for which the energy spectrum has the well-known − 5/3 scaling exponent.

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Section: Introductionmentioning

confidence: 60%

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

Kalmár‐Nagy

Bak

2019

Nonlinear Dyn

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“…The ratio of these intervals may vary significantly depending on the specific application. For example, following the ergodicity hypothesis [28,29], the flows with homogeneous directions can be averaged along these directions thus significantly reducing the time averaging interval. Opposite, the applications with complex geometries often need to perform much larger time averaging intervals to obtain reliable statistics compared to the time to reach the statistical equilibrium.…”

Section: Time-averaged and Ensemble-averaged Statisticsmentioning

confidence: 99%

An approach for accelerating incompressible turbulent flow simulations based on simultaneous modelling of multiple ensembles

Krasnopolsky

2018

Computer Physics Communications

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The present paper deals with the problem of improving the efficiency of large scale turbulent flow simulations. The high-fidelity methods for modelling turbulent flows become available for a wider range of applications thanks to the constant growth of the supercomputers performance, however, they are still unattainable for lots of real-life problems. The key shortcoming of these methods is related to the need of simulating a long time integration interval to collect reliable statistics, while the time integration process is inherently sequential.The novel approach with modelling of multiple flow states is discussed in the paper. The suggested numerical procedure allows to parallelize the integration in time by the cost of additional computations. Multiple realizations of the same turbulent flow are performed simultaneously. This allows to use more efficient implementations of numerical methods for solving systems of linear algebraic equations with multiple right-hand sides, operating with blocks of vectors. The simple theoretical estimate for the expected simulation speedup, accounting the penalty of additional computations and the linear solver performance improvement, is presented. The two problems of modelling turbulent flows in a plain channel and in a channel with a matrix of wall-mounted cubes are used to demonstrate the correctness of the proposed estimates and efficiency of the suggested approach as a whole. The simulation speedup by a factor of 2 is shown.

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“…Galanti and Tsinober (2004) proved that the turbulence, which is temporally steady and spatially homogeneous, is ergodic, but "partially turbulent flows" such as the mixed layer, wake flow, jet flow, flow around and boundary layer flow may be non-ergodic turbulence. However, it has been proven through atmospheric observational data that the turbulence ergodicity is related to the scale of turbulent eddies.…”

Section: Discussionmentioning

confidence: 99%

“…Afterward, a banausic ergodic theorem of stationary random processes was proven to provide a necessary and sufficient condition for the ergodicity of stationary random processes. Mattingly (2003) reviewed the research progress on ergodicity for stochastically forced Navier-Stokes equation, and that Galanti and Tsinober (2004) and Lennaert et al (2006) solved the Navier-Stokes equation by numerical simulation to prove that turbulence that is temporally steady and spatially homogeneous is ergodic. However, Galanti and Tsinober (2004) also indicated that such partially turbulent flows acting as mixed layer, wake flow, jet flow, flow around the boundary layer may be non-ergodic.…”

Section: Introductionmentioning

confidence: 99%

Ergodicity test of the eddy-covariance technique

Chen

Yinqiao

et al. 2015

Atmos. Chem. Phys.

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Abstract. The ergodic hypothesis is a basic hypothesis typically invoked in atmospheric surface layer (ASL) experiments. The ergodic theorem of stationary random processes is introduced to analyse and verify the ergodicity of atmospheric turbulence measured using the eddy-covariance technique with two sets of field observational data. The results show that the ergodicity of atmospheric turbulence in atmospheric boundary layer (ABL) is relative not only to the atmospheric stratification but also to the eddy scale of atmospheric turbulence. The eddies of atmospheric turbulence, of which the scale is smaller than the scale of the ABL (i.e. the spatial scale is less than 1000 m and temporal scale is shorter than 10 min), effectively satisfy the ergodic theorems. Under these restrictions, a finite time average can be used as a substitute for the ensemble average of atmospheric turbulence, whereas eddies that are larger than ABL scale dissatisfy the mean ergodic theorem. Consequently, when a finite time average is used to substitute for the ensemble average, the eddy-covariance technique incurs large errors due to the loss of low-frequency information associated with larger eddies. A multi-station observation is compared with a singlestation observation, and then the scope that satisfies the ergodic theorem is extended from scales smaller than the ABL, approximately 1000 m to scales greater than about 2000 m. Therefore, substituting the finite time average for the ensemble average of atmospheric turbulence is more faithfully approximate the actual values. Regardless of vertical velocity or temperature, the variance of eddies at different scales follows Monin-Obukhov similarity theory (MOST) better if the ergodic theorem can be satisfied; if not it deviates from MOST. The exploration of ergodicity in atmospheric turbulence is doubtlessly helpful in understanding the issues in atmospheric turbulent observations and provides a theoretical basis for overcoming related difficulties.

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Is turbulence ergodic?

Cited by 42 publications

References 6 publications

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

An intriguing analogy of Kolmogorov’s scaling law in a hierarchical mass–spring–damper model

An approach for accelerating incompressible turbulent flow simulations based on simultaneous modelling of multiple ensembles

Ergodicity test of the eddy-covariance technique

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