Estimates of the Relations between Eulerian and Lagrangian Scales in Large Reynolds Number Turbulence

Corrsin, S.

doi:10.1175/1520-0469(1963)020<0115:eotrbe>2.0.co;2

Cited by 174 publications

(88 citation statements)

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“…Figure 9 shows the spectra of the vertical velocity for a particle released in the middle of the CBL. As explained by Corssin (1963), the Lagrangian spectra in the inertial subrange follows a n Ϫ2 slope. With respect to the Eulerian one, the peak shifts toward smaller frequencies, as shown by Hanna (1981).…”

Section: Lagrangian Statisticsmentioning

confidence: 93%

Relating Eulerian and Lagrangian Statistics for the Turbulent Dispersion in the Atmospheric Convective Boundary Layer

Dosio

Arellano

Holtslag

et al. 2005

Journal of the Atmospheric Sciences

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Eulerian and Lagrangian statistics in the atmospheric convective boundary layer (CBL) are studied by means of large eddy simulation (LES). Spectra analysis is performed in both the Eulerian and Lagrangian frameworks, autocorrelations are calculated, and the integral length and time scales are derived. Eulerian statistics are calculated by means of spatial and temporal analysis in order to derive characteristic length and time scales. Taylor's hypothesis of frozen turbulence is investigated, and it is found to be satisfied in the simulated flow.Lagrangian statistics are derived by tracking the trajectories of numerous particles released at different heights in the turbulent flow. The relationship between Lagrangian properties (autocorrelation functions) and dispersion characteristics (particles' displacement) is studied through Taylor's diffusion relationship, with special emphasis on the difference between horizontal and vertical motion. Results show that for the horizontal motion, Taylor's relationship is satisfied. The vertical motion, however, is influenced by the inhomogeneity of the flow and limited by the ground and the capping inversion at the top of the CBL. The Lagrangian autocorrelation function, therefore, does not have an exponential shape, and consequently, the integral time scale is zero. If distinction is made between free and bounded motion, a better agreement between Taylor's relationship and the particles' vertical displacement is found.Relationships between Eulerian and Lagrangian frameworks are analyzed by calculating the ratio ␤ between Lagrangian and Eulerian time scales. Results show that the integral time scales are mainly constant with height for z/z i Ͻ 0.7. In the upper part of the CBL, the capping inversion transforms vertical motion into horizontal motion. As a result, the horizontal time scale increases with height, whereas the vertical one is reduced. Current parameterizations for the ratio between the Eulerian and Lagrangian time scales have been tested against the LES results showing satisfactory agreement at heights z/z i Ͻ 0.7.

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Section: Lagrangian Statisticsmentioning

confidence: 93%

Relating Eulerian and Lagrangian Statistics for the Turbulent Dispersion in the Atmospheric Convective Boundary Layer

Dosio

Arellano

Holtslag

et al. 2005

Journal of the Atmospheric Sciences

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“…Corrsin's conjecture (Corrsin 1963) states that, for |t − t ′ | large enough, (2.3) can be approximated by…”

Section: Turbulent Diffusion and Corrsin's Conjecturementioning

confidence: 99%

Turbulent diffusion in rapidly rotating flows with and without stable stratification

et al. 2004

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In this work, three different approaches are used for evaluating some Lagrangian properties of homogeneous turbulence containing anisotropy due to the application of a stable stratification and a solid-body rotation. The two external frequencies are the magnitude of the system vorticity 2Ω,chosen vertical here, and the Brunt-Väisälä frequency N,w h i c hg i ves the strength of the vertical stratification. Analytical results are derived using linear theory for the Eulerian velocity correlations (single-point, twotime) in the vertical and the horizontal directions, and Lagrangian ones are assumed to be equivalent, in agreement with an additional Corrsin assumption used by Kaneda (2000). They are compared with results from the kinematic simulation model (KS) by Nicolleau & Vassilicos (2000), which also incorporates the wave-vortex dynamics inherited from linear theory, and directly yields Lagrangian correlations as well as Eulerian ones. Finally, results from direct numerical simulations (DNS) are obtained and compared for the rotation-dominant case B =2Ω/N = 10, the stratificationdominant case B =1/10, the non-dispersive case B =1, and pure stratification B =0 and pure rotation N =0. The last situation is shown to be singular with respect to the mixed stratified/rotating ones. We address the question of the validity of Corrsin's simplified hypothesis, which states the equivalence between Eulerian and Lagrangian correlations. Vertical correlations are found to follow this postulate, but not the horizontal ones. Consequences for the vertical and horizontal one-particle dispersion are examined. In the analytical model, the squared excursion lengths are calculated by time integrating the Lagrangian (equal to the Eulerian) two-time correlations, according to Taylor's procedure. These quantities are directly computed from fluctuating trajectories by both KS and DNS. In the case of pure rotation, the analytical procedure allows us to relate Brownian t-asymptotic laws of dispersion in both the horizontal and vertical directions to the angular phase-mixing properties of the inertial waves. If stratification is present, the inertia-gravity wave dynamics, which affects the vertical motion, yields a suppressed vertical diffusivity, but not a suppressed horizontal diffusivity, since part of the horizontal velocity field escapes wavy motion.

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“…This poses no problems for the mean terms that appear in the transport equations (they are Eulerian), but the covariance terms are more complicated because they are in Lagrangian form. The problem of relating the Eulerian statistics that can be physically measured to the Lagrangian statistics that arise in transport theories is a central problem in hydrology and fluid dynamics [e.g., Corrsin, 1959Corrsin, , 1963 We offer the following example of how the Lagrangian correlation functions defined in (45a)-(45d) and (46a)-(46d) might be calculated numerically for a spatially nonstationary velocity field. As described above, one would begin by generating an ensemble of discretized conductivity fields with the same underlying statistical structure.…”

Section: Comment On the Specification Of The Covariance Functionsmentioning

confidence: 99%

Ensemble‐averaged equations for reactive transport in porous media under unsteady flow conditions

Wood¹,

Kawas²

1999

Water Resources Research

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Abstract. We present a method for deriving the ensemble-averaged reactive solute transport equation for unsteady, non-divergence-free flow field conditions. Our approach uses a cumulant expansion, Lie group theory, and time-ordered exponentials to develop the ensemble-averaged transport equation. The cumulant expansion is in powers of a arc, where a measures the magnitude of the perturbations of the transport and reaction operators and rc is the correlation time of these perturbations. Because the cumulant expansion avoids secular terms (terms in powers of time), the problem can be closed by rationally truncating the expansion. The truncated terms can be shown to be of lower order than those terms that are kept, provided that a particular constraint (in terms of the Kubo number) is met. The use of Lie group theory allows one to automatically combine the Eulerian and Lagrangian approaches. A particular time-ordered exponential that arises in the analysis can be interpreted as a translation operator that possesses a welldefined algebra. These translation operators appear in the second-order (covariance) terms of the cumulant expansion, and their effect is to shift one of the terms of the covariance functions relative to the other along the trajectory formed by the ensembleaveraged velocity field. This approach does not require neglecting the local dispersion tensor and has the advantage that no integral transformations are conducted; therefore all results are expressed in terms of.real space variables.

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Estimates of the Relations between Eulerian and Lagrangian Scales in Large Reynolds Number Turbulence

Cited by 174 publications

References 0 publications

Relating Eulerian and Lagrangian Statistics for the Turbulent Dispersion in the Atmospheric Convective Boundary Layer

Relating Eulerian and Lagrangian Statistics for the Turbulent Dispersion in the Atmospheric Convective Boundary Layer

Turbulent diffusion in rapidly rotating flows with and without stable stratification

Ensemble‐averaged equations for reactive transport in porous media under unsteady flow conditions

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