A comprehensive study is reported of the Lagrangian statistics of velocity, acceleration, dissipation and related quantities, in isotropic turbulence. High-resolution direct numerical simulations are performed on 643 and 1283 grids, resulting in Taylor-scale Reynolds numbers Rλ in the range 38-93. The low-wavenumber modes of the velocity field are forced so that the turbulence is statistically stationary. Using an accurate numerical scheme, of order 4000 fluid particles are tracked through the computed flow field, and hence time series of Lagrangian velocity and velocity gradients are obtained.The results reported include: velocity and acceleration autocorrelations and spectra; probability density functions (p.d.f.'s) and moments of Lagrangian velocity increments; and p.d.f.'s, correlation functions and spectra of dissipation and other velocity-gradient invariants. It is found that the acceleration variance (normalized by the Kolmogorov scales) increases as R½λ - a much stronger dependence than predicted by the refined Kolmogorov hypotheses. At small time lags, the Lagrangian velocity increments are distinctly non-Gaussian with, for example, flatness factors in excess of 10. The enstrophy (vorticity squared) is found to be more intermittent than dissipation, having a standard-deviation-to-mean ratio of about 1.5 (compared to 1.0 for dissipation). The acceleration vector rotates on a timescale about twice the Kolmogorov scale, while the timescales of acceleration magnitude, dissipation and enstrophy appear to scale with the Lagrangian velocity timescale.

▪ Abstract  A Lagrangian description of turbulence has unique physical advantages that are especially important in studies of mixing and dispersion. We focus on fundamental aspects, using mainly data from direct numerical simulations capable of great detail and precision when specific accuracy requirements are met. Differences between time evolution in Eulerian and Lagrangian frames illustrate the dominance of advective transport. We examine basic results in Kolmogorov similarity, giving an estimate of an inertial-range universal constant and the grid resolution and Reynolds number needed to attain the requisite scaling range of time lags. The Lagrangian statistics of passive scalars are discussed in view of current efforts in model development, with differential diffusion between multiple scalars being characterized by shorter timescales. We also note the need for new data in more complex flows and in other applications where a Lagrangian viewpoint is especially useful.

Existing experimental and numerical data suggest that the turbulence energy dissipation and enstrophy ͑i.e., the square of vorticity͒ possess different scaling properties, while available theory suggests that there should be no differences at sufficiently high Reynolds numbers. We have performed a series of direct numerical simulations with up to 2048 3 grid points where advanced computational power is used to increase the Reynolds number ͑up to 650 on the Taylor scale͒ or to resolve the small scales better ͑down to 1 / 4 of a Kolmogorov scale͒. Our primary goal is to assess the differences and similarities between dissipation and enstrophy. Special attention is paid to the effects of small-scale resolution on the quality and reliability of the data, in view of recent theoretical work ͓V. Yakhot and K. R. Sreenivasan, "Anomalous scaling of structure functions and dynamic constraints on turbulence simulations," J. Stat. Phys. 121, 823 ͑2005͔͒ which stipulates the resolution needed to obtain a moment of a given order. We also provide error estimates as a function of small-scale resolution. Probability density functions of dissipation and enstrophy at high Reynolds number reveal the presence of extreme events several thousands times of the mean. The extreme events in dissipation and enstrophy fields appear to scale alike, substantially overlap in space, and are nearly statistically isotropic, while fluctuations of moderate amplitudes, at least for the present Reynolds numbers, show persistent differences. Conditional sampling shows that intense dissipation is likely to be accompanied by similarly intense enstrophy, but intense enstrophy is not always accompanied by intense dissipation.

The scaling properties of one- and two-point statistics of the acceleration, pressure, and pressure gradient are studied in incompressible isotropic turbulence by direct numerical simulation. Ensemble-averaged Taylor-scale Reynolds numbers (Rλ) are up to about 230 on grids from 323 to 5123. From about Rλ 40 onwards the acceleration variance normalized by Kolmogorov variables is found to increase as Rλ1/2. This nonuniversal behavior is traced to the dominant irrotational pressure gradient contributions to the acceleration (whereas the much weaker solenoidal viscous part is universal). Longitudinal and transverse two-point correlations of the pressure gradient differ according to kinematic constraints, but both (especially the latter) extend over distances of intermediate scale size large compared to the Kolmogorov scale. These extended-range properties essentially provide the Eulerian mechanism whereby (as found in recent work) the accelerations of a pair of fluid particles can remain significantly correlated for relatively long periods of time even as they move apart from each other. Although a limited inertial range is attained in the energy spectrum, little evidence for classical inertial scaling is found in acceleration correlations and pressure structure functions. The probability density function (PDF) of pressure fluctuations has negatively skewed tails that exhibit a stretched-exponential form. Pressure gradient statistics show a rapid increase in intermittency with Reynolds number, characterized by widening tails in the PDF and large flatness factors. The practicality of computing acceleration correlations from velocity structure functions is also assessed using direct numerical simulations (DNS); within some resolution limitations good agreement is obtained with experimental data in grid turbulence at comparable Reynolds number.

Fully turbulent flows are characterized by intermittent formation of very localized and intense velocity gradients. These gradients can be orders of magnitude larger than their typical value and lead to many unique properties of turbulence. Using direct numerical simulations of the Navier-Stokes equations with unprecedented small-scale resolution, we characterize such extreme events over a significant range of turbulence intensities, parameterized by the Taylor-scale Reynolds number ( l R ). Remarkably, we find the strongest velocity gradients to empirically scale as t l