A second-order autoregressive equation is used to model the acceleration of fluid particles in turbulence in order to study the effect of Reynolds number on Lagrangian turbulence statistics. It is shown that this approach provides a good representation of dissipation subrange structure of Lagrangian velocity and acceleration statistics. The parameters of the model, two time scales representing the energy-containing and dissipation scales, are determined by matching the model velocity autocorrelation function to Kolmogorov similarity forms in the inertial subrange and the dissipation subrange. The model is tested against the Lagrangian statistics obtained by Yeung and Pope [J. Fluid Mech. 207, 531 (1989)] from direct numerical simulations of turbulence. Agreement between the model predictions and simulation data for second-order Lagrangian statistics such as the velocity structure function, the acceleration correlation function, and the dispersion of fluid particles is excellent, indicating that the main departures from Kolmogorov’s theory of local isotropy shown by the simulation data are due to low Reynolds number. For Reynolds numbers typical of laboratory experiments and direct numerical simulations of turbulence the root-mean-square dispersion of marked particles is changed from the Langevin equation (i.e., infinite Reynolds number) prediction by up to about 50% at large times. Most of this change can be accounted for by the change in the Lagrangian integral time scale. It is also shown that Reynolds number effects in laboratory dispersion or Lagrangian turbulence measurements can cause significant errors (typically of order 50%) when the value of the Kolmogorov Lagrangian structure function constant C0 is estimated by fitting the predictions of the Langevin equation to these data. A value C0 = 7 is obtained by fitting the new model to the direct simulation data.
▪ Abstract This review begins with the classical foundations of relative dispersion in Kolmogorov's similarity scaling. Analysis of the special cases of isotropic and homogeneous scalar fields is then used to establish most simply the connection with turbulent mixing. The importance of the two-particle acceleration covariance in relative dispersion is demonstrated from the kinematics of the motion of particle-pairs. A summary of the development of two-particle Lagrangian stochastic models is given, with emphasis on the assumptions and constraints involved, and on predictions of the scalar variance field for inhomogeneous sources. Two-point closures and kinematic simulation are also reviewed in the context of their prediction of the Richardson constant and other fundamental constants. In the absence of reliable field data, direct numerical simulations and laboratory measurements seem most likely to provide suitable data with which to test the assumptions and predictions of these theories.
We review the theoretical basis for, and the advantages of, random flight models for the trajectories of tracer particles in turbulence. We then survey their application to calculate dispersion in the principal types of atmospheric turbulence (stratified, vertically-inhomogeneous, Gaussian or non-Gaussian turbulence in the surface layer and above), and show that they are especially suitable for some problems (e.g., quantifying ground emissions).
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