Two-dimensional flow of a surface-active monolayer on a thin viscous film is considered. Simplifications of negligible gravity and pressure forces are made. Interfacial properties are described by simple model equations of state. Solutions are obtained for when the monolayer is scraped along the interface by a barrier and a steady state exists where surface advection is balanced by surface diffusion. Surface velocity, film thickness and spreading rate dependence on surface diffusivity are examined.

A family of Lagrangian stochastic models for the joint motion of particle pairs in isotropic homogeneous stationary turbulence is considered. The Markov assumption and well-mixed criterion of Thomson (1990) are used, and the models have quadratic-form functions of velocity for the particle accelerations. Two constraints are derived which formally require that the correct one-particle statistics are obtained by the models. These constraints involve the Eulerian expectation of the ‘acceleration’ of a fluid particle with conditioned instantaneous velocity, given either at the particle, or at some other particle's position. The Navier-Stokes equations, with Gaussian Eulerian probability distributions, are shown to give quadratic-form conditional accelerations, and models which satisfy these two constraints are found. Dispersion calculations show that the constraints do not always guarantee good one-particle statistics, but it is possible to select a constrained model that does. Thomson's model has good one-particle statistics, but is shown to have unphysical conditional accelerations. Comparisons of relative dispersion for the models are made.

In this paper we study acceleration statistics from laboratory measurements and direct numerical simulations in three-dimensional turbulence at Taylor-scale Reynolds numbers ranging from 38 to 1000. Using existing data, we show that at present it is not possible to infer the precise behavior of the unconditional acceleration variance in the large Reynolds number limit, since empirical functions satisfying both the Kolmogorov and refined Kolmogorov theories appear to fit the data equally well. We also present entirely new data for the acceleration covariance conditioned on the velocity, showing that these conditional statistics are strong functions of velocity, but that when scaled by the unconditional variance they are only weakly dependent on Reynolds number. For large values of the magnitude u of the conditioning velocity we speculate that the conditional covariance behaves like u 6 and show that this is qualitatively consistent with the stretched exponential tails of the unconditional acceleration probability density function ͑pdf͒. The conditional pdf is almost identical in shape to the unconditional pdf. From these conditional covariance data, we are able to calculate the conditional mean rate of change of the acceleration, and show that it is consistent with the drift term in second-order Lagrangian stochastic models of turbulent transport. We also calculate the correlation between the square of the acceleration and the square of the velocity, showing that it is small but not negligible.

The relative dispersion of fluid particle pairs in isotropic turbulence is studied using direct numerical simulation, in greater detail and covering a wider Reynolds number range than previously reported. A primary motivation is to provide an important resource for stochastic modelling incorporating information on Reynolds-number dependence. Detailed results are obtained for particle-pair initial separations from less than one Kolmogorov length scale to larger than one integral length scale, and for Taylor-scale Reynolds numbers from about 38 to 230. Attention is given to several sources of uncertainty, including sample size requirements, value of the one-particle Lagrangian Kolmogorov constant, and the temporal variability of space-averaged quantities in statistically stationary turbulence.Relative dispersion is analysed in terms of the evolution of the magnitude and angular orientation of the two-particle separation vector. Early-time statistics are consistent with the Eulerian spatial structure of the flow, whereas the large-time behaviour is consistent with particle pairs far apart moving independently. However, at intermediate times of order several Kolmogorov time scales, and especially for small initial separation and higher Reynolds numbers, both the separation distance and its rate of change (called the separation speed) are highly intermittent, with flatness factors much higher than those of Eulerian velocity differences in space. This strong intermittency is a consequence of relative dispersion being affected by a wide range of length scales in the turbulent flow as some particle pairs drift relatively far apart. Numerical evidence shows that substantial dispersion occurs in the plane orthogonal to the initial separation vector, which implies that the orientation of this vector has, especially for small initial separation, only limited importance.

We use Lagrangian stochastic models and direct numerical simulation for stationary isotropic turbulence to calculate backwards relative dispersion statistics, that is, the statistics of the earlier locations of particle pairs that at a later time are located at prescribed locations. We find that, in general, backwards relative dispersion proceeds at a much faster rate than relative dispersion forwards in time, and the difference between the two is sensitive to the nature of the flow field. The difference vanishes for Gaussian flows and for white-noise (in time) flows (for which relative dispersion can be described by a diffusion equation), suggesting that theories such as two-point closure and kinematic simulation do not differentiate between backwards and forwards dispersion. Backwards relative dispersion is very sensitive to the details of the tails of the probability density function for the Eulerian velocity difference between two points.