Two-particle Dispersion Research Articles

Lagrangian statistics in fully developed turbulence

The statistics of Lagrangian particles transported by a three-dimensional fully developed turbulent flow is investigated by means of high-resolution direct numerical simulations. The analysis of single trajectories reveals the existence of strong trapping events vortices at the Kolmogorov scale which contaminates inertial range statistics up to 10 τη. For larger time separations, we find that Lagrangian structure functions display intermittency in agreement with the prediction of the multifractal model of turbulence. The study of two-particle dispersion shows that the probability density function of pair separation is very close to the original prediction of Richardson of 1926. Nevertheless, moments of relative dispersion are strongly affected by finite Reynolds effects, thus limiting the possibility to measure numerical prefactors, such as the Richardson constant g. We show how, by using an exit time statistics, it is possible to have a precise estimation of g which is consistent with recent laboratory measurements.

Relative dispersion in isotropic turbulence. Part 2. A new stochastic model with Reynolds-number dependence

A new model for Lagrangian particle-pair separation in turbulent flows is developed and compared with data from direct numerical simulations (DNS) of isotropic turbulence. The model formulation emphasizes (i) non-Gaussian behaviour in Eulerian and Lagrangian statistics, (ii) the occurrence of large separation velocities, (iii) the role of straining and streaming flow structure as recognized in kinematic simulations of turbulence, and (iv) the role of conditionally averaged accelerations in stochastic modelling of turbulent relative dispersion. Previous stochastic models of relative dispersion have produced unrealistic behaviour, particularly in the dissipation subrange where viscous effects are important, which have led to questions on the adequacy of stochastic modelling. However, this failure can now be recognized as inadequate detail in formulation, which is explained and rectified in this paper. The model is quasi-one-dimensional in nature, and is focused on the statistics of particle-pair separation distance and its rate of change, referred to as the separation speed. Detailed comparisons are presented at several Reynolds numbers using the DNS database reported in a companion paper (Part 1). Up to fourth-order moments for these quantities are examined, as well as the separation-distance probability density function, which is discussed in the context of recent claims of Richardson scaling in the literature. The model is able to account for the spatial representation of straining regions as well as incompressibility of the flow, and is shown to reproduce strong non-Gaussianity and intermittency in the Lagrangian separation statistics observed in DNS. Comparisons with recent physical experiments are also remarkably consistent. This work demonstrates that stochastic models when properly formulated are effective and efficient representations of the dispersion process and this general class of models therefore possess great utility for calculations of both one-particle and two-particle dispersion. The techniques developed in this paper will facilitate such general model development.

Two-Particle Dispersion in Superfluid Turbulence

Recent experiments have shown that superfluid turbulence is similar to ordinary classical turbulence with respect to temporal decay and energy spectra. Since superfluid vorticity is concentrated to thin filaments, superfluid turbulence is geometrically very different from ordinary turbulence, in which eddies can be of any size and strength. In order to explore differences and similarities between the two forms of turbulence, we investigate and compare two-particle dispersion, a property which quantifies the degree of divergence of turbulent flow trajectories.

A Simple Relative Dispersion Model for Concentration Fluctuations in Contaminant Clouds

The relative dispersion process for clouds of contaminant in generic atmospheric flow is considered. The properties of the separation distance for pairs of particles are simplified by implicitly averaging over the spatial domain of the dispersing cloud. Representative statistics and simplified sets of measurements for characterizing two-particle dispersion in complex flows are identified. A Lagrangian stochastic model of relative dispersion equivalent to processes in homogeneous and isotropic turbulence at high Reynolds numbers is derived. The model uses a new formulation for parameterizing the acceleration of separation, satisfies the criterion of conserving a well-mixed distribution of particle separations, and accounts explicitly for non-Gaussian statistics of the turbulence velocity differences. The results are in very good agreement with similarity theory in the inertial range and are consistent with uncorrelated velocities at length scales larger than the turbulence integral scale. The model is applied to the estimation of fluctuating concentration fields, which is relevant for representing the relative dispersion part of popular meandering plume and puff approaches. The dependence of mean-square concentration and concentration fluctuations on the source size is eliminated via a new scaling law for the time, which in fact determines a universal behavior for the concentration field. Simple formulas are derived that are consistent with previous theories, and they are successfully tested against numerical simulations.

Statistics of two-particle dispersion in two-dimensional turbulence

We investigate Lagrangian relative dispersion in direct numerical simulation of two-dimensional inverse cascade turbulence. The analysis is performed by using both standard fixed time statistics and an exit time approach. The latter allows a more precise determination of the Richardson constant which is found to be g≃4 with a possible weak finite-size dependence. Our results show only small deviations with respect to the original Richardson’s description in terms of diffusion equation. These deviations are associated with the long-range correlated nature of the particles’ relative motion. The correlation, or persistence, parameter is measured by means of a Lagrangian “turning point” statistics.

Two-particle dispersion in model two-dimensional velocity fields

We consider two-particle dispersion in a velocity field, where the relative two-point velocity scales according to v2(r) ∝rα and the corresponding correlation time scales as τ(r) ∝rβ, and fix α = 2/3, as typical for turbulent flows. We show that two generic types of dispersion behavior arize: For α/2 + β 1, the relative motion is strongly correlated. The case of Kolmogorov flows corresponds to a marginal, nongeneric situation. In this case, depending on the particular parameters of the flow, the dispersion behavior can be rather diffusive or rather ballistic.

Particle dispersion in synthetic turbulent flows

We study particle dispersion advected by a synthetic turbulent flow from a Lagrangian perspective and focus on the two-particle and cluster dispersion by the flow. It has been recently reported that Richardson's law for the two-particle dispersion can stem from different dispersion mechanisms, and can be dominated by either diffusive or ballistic events. The nature of the Richardson dispersion depends on the parameters of our flow and is discussed in terms of the values of a persistence parameter expressing the relative importance of the two above-mentioned mechanisms. We support this analysis by studying the distribution of interparticle distances, the relative velocity correlation functions, as well as the relative trajectories.

Ballistic versus diffusive pair dispersion in the Richardson regime

Two-particle dispersion in fully developed turbulence is modeled within an asymmetric L\'evy-walk framework which yields the Richardson ${t}^{3}$ law and which recovers recently observed results on the probability distribution function of pair separations in two-dimensional turbulence [M.C. Jullien, J. Paret, and P. Tabeling, Phys. Rev. Lett. 82, 2872 (1999)]. The ballistic nature of the Richardson dispersion is discussed in terms of the persistence parameter and compared with the Richardson diffusive behavior.

Two-particle dispersion by correlated random velocity fields.

We consider the two-particle dispersion in a velocity field, where the relative two-point velocity scales according to v(2)(r) proportional, variantr(alpha) and the corresponding correlation time scales as tau(r) proportional r(beta). We show that for alpha/2+beta<1 the diffusion approximation holds, and the increase in the interparticle distances is governed by the distance-dependent diffusion coefficient K(r) proportional r(alpha+beta). The possible regimes outside of the validity of diffusion approximation are discussed. The Kolmogorov scaling in turbulent flow alpha=beta=2/3 corresponds to a borderline situation. The experimental data for this case suggest that the separation regime is probably ballistic.

A Lagrangian model for turbulent dispersion with turbulent-like flow structure: Comparison with direct numerical simulation for two-particle statistics

A wide range of relative two-particle dispersion statistics from the Lagrangian Kinematic Simulation (KS) model, which contains turbulent-like flow structures, compares well with Yeung’s [Phys. Fluids 6, 3416 (1994)] DNS results. In particular, the Lagrangian flatness factor μ4(t) compares excellently (better than Heppe’s [J. Fluid Mech. 357, 167 (1998)] nonlinear stochastic model). For higher Reynolds numbers the results from KS show that μ4(t) is significantly greater than 3 over a wide range of times within the inertial range of time scales.

Lagrangian and Tracer Evolution in the Vicinity of an Unstable Jet

The dynamics of Lagrangian particles and tracers in the vicinity of a baroclinically unstable zonal jet are investigated in a simple two-layer model with an initially quiescent lower layer. The presence of a growing wave induces a particle drift dominated by Stokes drift rather then the contribution of the wave to the mean Eulerian velocity. Stable and unstable waves have zonal Stokes drift with similar meridional structure while only unstable waves possess meridional drift, which is in the direction of increasing meridional wave displacement. Particle dispersion in the upper layer is maximum at critical lines, where the jet and phase speeds are equal. In the lower layer, dispersion is maximum where the wave amplitude is maximum. Zonal mean tracer evolution is formulated as an advection‐diffusion equation with an order Rossby number advection and an orderone eddy diffusion. The latter is proportional to two-particle dispersion. Finite amplitude simulations of the flow reveal that small amplitude theory has predictive value beyond the range for which it is strictly valid. Mixing (as opposed to stirring) is maximum near cat’s-eye-like recirculation regions at the critical lines. In the lower layer the pattern of convergence and divergence of the flow locally increases tracer gradients, resulting in stirring yet with a much slower mixing rate than in the upper layer. Meridional eddy diffusion (or particle dispersion) alone is not sufficient for prediction of mixing intensity. Rotation, which is quantified by the cross-correlation of meridional and zonal displacements, must also be present for mixing. These results are consistent with observations of tracer and floats in the vicinity of the Gulf Stream.

Two-particle dispersion in turbulentlike flows

Kinematic simulations are non-Markovian Lagrangian models of dispersion that incorporate turbulentlike flow structure. We investigate the conditions for two-particle dispersion to be local in a turbulentlike flow, and the dependence of the Richardson constant ${G}_{\ensuremath{\Delta}}$ on the topology of individual realizations of the flow.

A family of stochastic models for two-particle dispersion in isotropic homogeneous stationary turbulence

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.

The fractal dimension of drifter trajectories and estimates of horizontal eddy-diffusivity

We analyse drifter data from the Northeast Atlantic and find that single-particle motion has fractal dimension D = 1.28 ± 0.08 at scales 5 to 100 km. The single particle trajectories are modeled using fractional Brownian motion. Integral time scale is therefore shown to be a function of both the time between position fixes and the length of the total period the drifter was tracked. The variance (particle dispersion) grows as approximately t 3/2 . The probability density function of single-particle velocity is observed to be leptokurtic, consistent with the fractional Brownian motion model. The present work reports two-particle trajectories with fractal dimension D = 1.3 for scales from 8 km to 150 km. The two-particle dispersion is modelled as accelerated fractional Brownian motion, in which speed becomes a function of time elapsed since the particles were very near to each other. The acceleration is chosen to be consistent with the patch variance growing as t 2.34 . The dependence of relative velocity between pairs of drifters separated by a distance l , is observed to scale as l 0.3 . This is consistent with the accelerated fractional Brownian motion model for two-particle motion. DOI: 10.1034/j.1600-0870.1991.t01-1-00008.x

Turbulence and dispersion parameters derived from smoke-plume photoanalysis

Instantaneous and time-averaged photographs of a smoke plume generated in a large wind tunnel are analyzed to obtain downwind estimates of single-particle and two-particle dispersion rates. Single particle i.e. time-averaged, dispersion rates are measured directly from a methane gas plume released and sampled under wind tunnel conditions identical to the smoke-plume release. The correlation between smoke-plume derived dispersion rates and measured values is 0.99; the r.m.s. error between these values is 7.6 per cent, and the geometric mean of the ratio of measured to derived values is 1.09. Using estimates of single-particle and two-particle dispersion values, the following turbulence parameters are calculated: crosswind velocity variance, σ 2 v, Lagrangian integral time scale, τ L, eddy diffusivity, K, eddy dissipation rate, ε, and crosswind turbulence intensity, i. The values of these parameters agree well with values directly measured from the flow field. Details on the use of smoke-plume methods are given, and it is demonstrated that this method is accurate, useful and cost effective.

The relation between the Markov process theory and Kolmogoroff's theory of turbolence and the extension of Kolmogoroff's laws

By using the physical analysis described in Paper I (Part I of this paper), we shall establish, in a certain way, the quantitative relation between the Markov process theory of two particle dispersion in a turbulence of very large Reynolds number and the Kolmogoroff's theory. In terms of this relation and the results of two-particle dispersion, we shall obtain the structure functions, the correlation functions and the energy spectrum, which are applicable not only to the inertial subrange, but also to the whole range of the wave number less than that in the inertial subrange. The Kolmogoroff's “2/3 law” and “−5/3 Law” are the asymptotic cases of the present result for large k. Thus, the present resuit is an extension of Kolmogoroff's laws.

A stochastic model of two-particle dispersion and concentration fluctuations in homogeneous turbulence

A new definition of concentration fluctuations in turbulent flows is proposed. The definition implicitly incorporates smearing effects of molecular diffusion and instrumental averaging. A stochastic model of two-particle dispersion, consistent with this definition, is formulated. The stochastic model is an extension of Taylor's (1921) model and is consistent with Richardson's represents net destruction of fluctuations by relative dispersion. Only the first term is included in the usual one-particle model (Corrsin 1952).

Dispersion contribution of two-atom interaction energy: Multipole interactions

Gives a theory of non-retarded dispersion interaction energy of molecules with charge distributions on them that readily allows inclusion of higher-order multipole interactions. The two-particle dispersion energy for a pair of harmonic oscillators is evaluated for dipole-dipole, dipole-quadrupole and quadrupole-quadrupole interactions. It also is found that the dispersion energy remains finite for all separations between the oscillators, supporting the earlier result of Mahanty and Ninham (1975) for dipole-dipole interactions with distributed dipole moments.