Particle Pair Separation Research Articles

Mixing in fully chaotic flows

Passive scalar mixing in fully chaotic flows is usually explained in terms of Lyapunov exponents, i.e., rates of particle pair separations. We present a unified review of this approach (which encapsulates also other nonchaotic flows) and investigate its limitations. During the final stage of mixing, when the scalar variance decays exponentially, Lyapunov exponents can fail to describe the mixing process. The failure occurs when another mixing mechanism, first introduced by Fereday et al. [Phys. Rev. E 65, 035301 (2002)], leads to a slower decay than the mechanism based on Lyapunov exponents. Here we show that this mechanism is governed by the large-scale nonuniformities of the flow which are different from the small scale stretching properties of the flow that are captured by the Lyapunov exponents. However, during the initial stage of mixing, i.e., the stage when most of the scalar variance decays, Lyapunov exponents describe well the mixing process. We develop our theory for the incompressible and diffusive baker map, a simple example of a chaotic flow. Nevertheless, our results should be applicable to all chaotic flows.

The Mean Pairwise Peculiar Velocity in Cosmological [ITAL]N[/ITAL]-Body Simulations: Time Variation, Scale Dependence, and the Stable Condition

We report on the detailed analysis of the mean pairwise peculiar velocity profile in high-resolution cosmological N-body simulations (N = 8.8 × 106 particles in a sphere of 50-200 Mpc radius). In particular, we examine the validity and limitations of the stable condition, which states that the mean physical separation of particle pairs is constant on small scales. We find a significant time variation (an irregular oscillatory behavior) in the mean pairwise peculiar velocity in nonlinear regimes. We argue that this behavior is not due to any numerical artifact but is a natural consequence of the continuous merging processes in the hierarchical clustering universe. While such a time variation is significant in a relatively local patch of the universe, the global average over a huge spatial volume of ≳200 Mpc3 does not reveal any systematic departure from the stable condition. Thus, we conclude that the mean pairwise peculiar velocity is rather unstable statistically but that it still satisfies the stable condition when averaged over the cosmological volume.

A scalar subgrid model with flow structure for large-eddy simulations of scalar variances

A new model to simulate passive scalar fields in large-eddy simulations of turbulence is presented. The scalar field is described by clouds of tracer particles and the subgrid contribution of the tracer displacement is modelled by a kinematic model which obeys Kolmogorov's inertial-range scaling, is incompressible and incorporates turbulent-like flow structure of the turbulent small scales. This makes it possible to study the scalar variance field with inertial-range effects explicitly resolved by the kinematic subgrid field while the LES determines the value of the Lagrangian integral time scale TL. In this way, the modelling approach does not rely on unknown Lagrangian input parameters which determine the absolute value of the scalar variance.The mean separation of particle pairs displays a well-defined Richardson scaling in the inertial range, and we find that the Richardson constant GΔ ≈ 0.07 which is small compared to the value obtained from stochastic models with the same TL. The probability density function of the separation of particle pairs is found to be highly non-Gaussian in the inertial range of times and for long times becomes Gaussian. We compute the scalar variance field for an instantaneous line source and find good agreement with experimental data.

The Relative Dispersion of Particle Pairs in Stationary Homogeneous Turbulence

Abstract The relative dispersion of particle pairs in stationary homogeneous turbulence is investigated in numerical simulations using one-dimensional and quasi-one-dimensional Lagrangian stochastic models, satisfying the well-mixed condition for Gaussian and non-Gaussian two-point velocity distribution functions. Predictions for the statistics of particle-pair separations are shown to be sensitively dependent upon third and higher moments of the two-point velocity distribution function. The results of numerical simulations suggest that Lagrangian stochastic models based upon Gaussian two-point velocity distribution functions tend to underpredict the intensity of concentration fluctuations at small times.

Fluid particle dispersion in homogeneous turbulent shear flow

The dispersion of Lagrangian fluid particle pairs in homogeneous turbulent shear flow is studied by a direct numerical simulation using 256×1282 grid points. It is emphasized that anisotropy requires the effects of both magnitude and orientation of the initial separation vector on two-particle statistics to be considered separately, especially at early times when these statistics are primarily determined by the Eulerian structure of the velocity gradient fluctuations. The effects of initial particle positions and pair separation on displacement and dispersion statistics are examined in relation to classical theory. Particle-pair dispersion is found to be most effective in the streamwise direction, and especially so for particle pairs initially separated in the direction of the mean velocity gradient. Similarly, two-particle velocity correlations are anisotropic, being strongest in the streamwise component but weakest in the cross-stream component. The particle-pair separation distance probability density function differs significantly from a Gaussian form used in most dispersion models for air-quality applications. It is hoped that the new results will be helpful for extending two-particle stochastic models to anisotropic turbulent flows.

One- and two-particle Lagrangian acceleration correlations in numerically simulated homogeneous turbulence

Lagrangian statistics of the fluid particle acceleration are studied by direct numerical simulation, in stationary isotropic turbulence (at three different Reynolds numbers) and homogeneous shear flow with uniform mean shear rate. The one-particle acceleration autocorrelation decays rapidly with time, with a zero crossing just over two Kolmogorov time scales. In contrast, two-particle correlations are relatively persistent, especially for particle pairs of small initial separation distance. Results for intermediate times at a Taylor-scale Reynolds number of 140 resemble a t−1 inertial-range scaling suggested in the literature, but even higher Reynolds numbers are needed for more definitive comparisons. Use of a theoretical argument and conditional sampling indicates that the two-particle correlation is determined by a coupling between a correlation localized in space and a particle-pair separation probability density of positive skewness. The scenario which emerges is that whereas some fluid particles accelerate rapidly away from each other, the majority of particle pairs can still be relatively close together and hence help maintain the two-particle correlation at significant levels. Acceleration correlations in homogeneous shear flow are found to display a tendency toward local isotropy.

Diffusion in stably stratified turbulence

We examine results of direct numerical simulations (DNS) of homogeneous turbulence in the presence of stable stratification. We focus on the effects of stratification on eddy diffusion, and the distribution of pairs of particles released in the flow. DNS results are presented over a range of stratification, and at Reynolds numbers compatible with aliased free spectral results for a resolution of 128 mesh points. We compare results for particle dispersion to simple analytic theories such as that proposed by Csanady (1964) and Pearson et al. (1983) by adapting the basic Langevin model to decaying turbulence at low Reynolds numbers. Stable stratification is found to arrest both single particle displacements and pair separation in the direction of stratification, but it leaves these quantities nearly unaltered in the transverse direction. With respect to the dynamics of stratified flows, we find that regions of strong viscous dissipation are intermittently spaced, and are associated with large horizontal vorticity, consistent with recent experimental results by Fincham et al. (1994).

The separation of particle pairs in the eddy-damped quasinormal Markovian approximation

The implications of the eddy-damped quasinormal Markovian (EDQNM) approximation for the separation of particle pairs in (three-dimensional) inertial subrange turbulence are explored. In particular the EDQNM equations are solved for the particle separation probability distribution, without making the usual approximation of replacing the damping time scales by their large time asymptotic form. A number of properties of the separation probability distribution that are of significance for the prediction of concentration fluctuations are calculated. The values show quite large departures from the values obtained with the large time form of the damping time scales.

Digital particle image velocimetry

Digital particle image velocimetry (DPIV) is the digital counterpart of conventional laser speckle velocitmetry (LSV) and particle image velocimetry (PIV) techniques. In this novel, two-dimensional technique, digitally recorded video images are analyzed computationally, removing both the photographic and opto-mechanical processing steps inherent to PIV and LSV. The directional ambiguity generally associated with PIV and LSV is resolved by implementing local spatial cross-correlations between two sequential single-exposed particle images. The images are recorded at video rate (30 Hz or slower) which currently limits the application of the technique to low speed flows until digital, high resolution video systems with higher framing rates become more economically feasible. Sequential imaging makes it possible to study unsteady phenomena like the temporal evolution of a vortex ring described in this paper. The spatial velocity measurements are compared with data obtained by direct measurement of the separation of individual particle pairs. Recovered velocity data are used to compute the spatial and temporal vorticity distribution and the circulation of the vortex ring.

A Lagrangian analysis of turbulent diffusion

This is an analysis of diffusion of a scalar field by molecular transport and isotropic turbulence. Existing results are surveyed, and some new results are advanced. The discussion is supported with oceanographic and atmospheric observations of dispersion and diffusion. The existing results were originally obtained using a variety of mathematical techniques. However, all results are derived here using an approximate solution of the Lagrangian form of the advection‐diffusion equation. The approximation is equivalent to neglecting the spatial dependence of the transformation factors in the Lagrangian representation of the molecular flux divergence. Examinations of the diffusive subranges show the approximation to justified: infinitesimal line stretching is either controlled by relatively large scale shears (viscous‐diffusive subrange at large Prandtl number) or else is negligible during the diffusion process (inertia‐diffusive subrange at small Prandtl number). Estimation of scalar mean fields, total variances, and wave number spectra requires, in general, joint statistics of infinitesimal line stretching and either single particle displacement or particle pair separation. Normality is assumed for displacement statistics; separation statistics are determined from the Richardson‐Kraichnan equation. A simple derivation of that equation is presented here. Joint stretching‐separation statistics are modeled by a uniform shear flow, with time‐dependent amplitudes described by the Wiener process (white noise). With the possible exception of this random process, the only mathematics required here is elementary calculus, so details have been kept to a minimum. In the diffusion problems considered here, the turbulence is isotropic. However, both the approximate solution of the advection‐diffusion equation and the equations for joint displacements are equally valid for inhomogeneous turbulence.

The tensile strength of powders

A theory of the tensile strength of powders, incorporating the effect of powder density, particle size distribution and interparticle force, has been developed. It enables the experimental results for different size fractions of each of five materials of different chemical and physical nature to be correlated with the particle size distribution. The large variation of tensile strength with small changes in density is interpreted to be a manifestation of the dependence of the interparticle force with surface separation of particle-pairs. A law of corresponding states is also established and it correlates the tensile strength data of the five materials. It is suggested that the functional form of the interparticle forces is very nearly the same for the five materials. An estimate is made of the energy and length parameters that characterise the interparticle forces involved. The practical use of the correlations obtained for the prediction of tensile strength from particle size distribution determination alone is discussed.