Trajectories Of Tracer Particles Research Articles

On The Dynamical Content Of Lagrangian Stochastic Models In The Well-Mixed Class

It is shown how the correspondence between Lagrangian stochasticmodels and second-moment closures of the scalar-flux equation can be exploited to distinguishbetween Lagrangian stochastic models in the well-mixed class. It is found that physically realisticclosures of the scalar-flux equation correspond to Lagrangian stochastic models that have non-zero`spin' and so produce spiralling tracer-particle trajectories, whilst `zero-spin'models correspond to the isotropic-production model of scalar-fluxes.Lagrangian stochastic models consistent with rapid distortion theory and Speziale's transformation rule for the Reynolds stressequations in the extreme limit of two-dimensional turbulence are also shown to have non-zero spin.The residual non-uniqueness associated with satisfaction of thewell-mixed condition and the specification of mean spin is shown to be related to the helicity oftracer-particle trajectories. Investigations are also made of the influence upon turbulent dispersion oftime-dependent spin and of mean rotations of the fluctuating Lagrangian acceleration vector(i.e., second-order spin).

Effects of periodicity on flow and dispersion through closely packed fixed beds of spheres.

A lattice-Boltzmann formulation is used to investigate the effects of "periodicity" (geometry) on fluid flow and tracer-particle dispersion through fixed beds of spheres comprising of closely packed layers. In the "period-1" arrangement, spheres in the adjacent layers contact at their poles while the "period-2" and "period-3" arrangements correspond to hexagonal and faced-centered cubic close packing. For all three packing arrangements, there is a transition with increasing Reynolds number from a power law to a log-normal distribution of kinetic energies and, velocity and vorticity become more closely aligned giving rise to helical tracer-particle trajectories. It is suggested that these flow characteristics, unlike the stability of flow and the distribution of helicity, are largely insensitive to geometry, even when the geometry creates direct channels through the pack bed orientated along the gradient in applied pressure. For steady flows and strongly turbulent flows, such channels are predicted to provide direct routes for dispersion through a packed bed, while for weakly turbulent flows they influence dispersion primarily by destabilizing the flow and thereby promoting dispersion throughout a bed. The dispersion of tracer-particles released from a source located on or close to a "stagnation streamline" is predicted to be faster than ballistic in the near field and the transition to long-time Fickian diffusion is predicted to be distinguished by a regime of subdiffusion.

Electroviscous phenomena in colloidal dispersions

Electroviscous phenomena arise from a coupling of hydrodynamics and electrokinetics. This coupling gives rise to a number of interesting phenomena, such as an ion-size dependence of (i) diffusion coefficients of colloidal particles, and (ii) coagulation and deposition rates, interesting trajectories of tracer particles around large charged particles, an increase in viscosity and electrokinetic lift. The origins of these phenomena are being discussed.

Transport and mixing in Stokes flow: the effect of chaotic dynamics on the blinking stokeslet

Mixing and transport processes associated with slow viscous flows are studied in the context of a blinking stokeslet above a plane rigid boundary. Whilst the motivation for this study comes from feeding currents due to cilia or flagella in sessile micro- organisms, other applications in physiological fluid mechanics where eddying motions occur include the enhanced mixing which may arise in ‘bolus’ flow between red blood cells, peristaltic motion and airflow in alveoli. There will also be further applications to micro-engineering flows at micron lengthscales. This study is therefore of generic interest because it analyses the opportunities for enhanced transport and mixing in a Stokes flow environment in which one or more eddies are a central feature.The central premise in this study is that the flow induced by the beating of microscopic flagella or cilia can be modelled by point forces. The resulting system is mimicked by using an implicit map, the introduction of which greatly aids the study of the system's dynamics. In an earlier study, Blake & Otto (1996), it was noticed that the blinking stokeslet system can have a chaotic structure. Poincaré sections and local Lyapunov exponents are used here to explore the structure of the system and to give quantitative descriptions of mixing; calculations of the barriers to diffusion are also presented. Comparisons are made between the results of these approaches. We consider the trajectories of tracer particles whose density may differ from the ambient fluid; this implies that the motion of the particles is influenced by inertia. The smoothing effect of molecular diffusion can be incorporated via the direct solution of an advection–diffusion equation or equivalently the inclusion of white noise in the map. The enhancement to mixing, and the consequent ramifications for filter feeding due to chaotic advection are demonstrated.

Flow and dispersion through a close-packed fixed bed of spheres

Fluid flow through a close-packed fixed bed of spheres in a face-centered cubic arrangement is investigated in numerical simulations using a lattice-Boltzmann formulation. The dispersion of a tracer gas is studied both experimentally and numerically. At low Reynolds numbers, Re</=14, the flow is steady with a distribution of normalized local kinetic energies that follows a power law over roughly three orders of magnitude. Consequently the "stagnant" zones play a significant role in determining transport through the packed bed in contrast with the dangling ends for the analogous electrical transport problem, where the distribution of local currents is log binomial. At higher Reynolds numbers transitions to time-oscillatory and chaotically (turbulently) varying flows are predicted to occur with a crossover to a log-normal distribution of local kinetic energies. At the onset of transverse velocity fluctuations the simulated trajectories of tracer particles can cross planes of symmetry, resulting in an abrupt enhancement of dispersion. The dispersion of tracer particles in the time-oscillatory and chaotic varying flows is predicted to be Fickian in the far field. Model predictions for dispersion in a chaotic flow with Re approximately 100 are shown to be in good agreement with experiment.

On the non-uniqueness of Lagrangian stochastic models

Lagrangian stochastic (LS) models are used to predict mean concentrations of tracer particles within a mechanically ventilated building where the turbulent flow is three-dimensional and inhomogeneous. The models satisfy the well-mixed condition for Gaussian turbulence but produce different statistics for increments d θ k ′ to the orientation, θ k ′, of the Lagrangian velocity fluctuation vector. Of the models tested, the best agreement with the experimental data is obtained with Thomson’s (1987, J. Fluid. Mech. 180, 529–556) model, which is effectively a “zero-spin” model, having 〈 dθ k′; x〉 ≈ 0 . Mean concentrations of tracer-particles are less well predicted by the “zero-spin” model of Flesch and Wilson (1992), Boundary Layer Meteorol. 61, 349–374), generalized to three dimensions, and by models having 〈 dθ k′; x〉 ≠ 0 which produce tracer-particle trajectories with a preferred handedness. It is suggested that the specification of 〈 dθ k′; x〉 is not sufficient to distinguish the better LS models within the well-mixed class.

An efficient Lagrangian stochastic model of vertical dispersion in the convective boundary layer

We consider the one-dimensional case of vertical dispersion in the convective boundary layer (CBL) assuming that the turbulence field is stationary and horizontally homogeneous. The dispersion process is simulated by following Lagrangian trajectories of many independent tracer particles in the turbulent flow field, leading to a prediction of the mean concentration. The particle acceleration is determined using a stochastic differential equation, assuming that the joint evolution of the particle velocity and position is a Markov process. The equation consists of a deterministic term and a random term. While the formulation is standard, attention has been focused in recent years on various ways of calculating the deterministic term using the well-mixed condition incorporating the Fokker–Planck equation. Here we propose a simple parameterisation for the deterministic acceleration term by approximating it as a quadratic function of velocity. Such a function is shown to represent well the acceleration under moderate velocity skewness conditions observed in the CBL. The coefficients in the quadratic form are determined in terms of given turbulence statistics by directly integrating the Fokker–Planck equation. An advantage of this approach is that, unlike in existing Lagrangian stochastic models for the CBL, the use of the turbulence statistics up to the fourth order can be made without assuming any predefined form for the probability distribution function (PDF) of the velocity. The main strength of the model, however, lies in its simplicity and computational efficiency. The dispersion results obtained from the new model are compared with existing laboratory data as well as with those obtained from a more complex Lagrangian model in which the deterministic acceleration term is based on a bi-Gaussian velocity PDF. The comparison shows that the new model performs well.

Investigation on particle trajectories and lagrangian statistics at the outlet of a circular jet

Abstract The analysis of the near flow field of a circular turbulent jet at small Reynolds numbers is performed by Particle Tracking Velocimetry (PTV). This measurement technique allows the tracer particle trajectories to be followed and the Lagrangian statistics to be computed. From Lagrangian correlation coefficients mean square displacements (MSD) of flow particles on longitudinal and vertical cross-sections have been determined. The main results are the following: (i) particles at the centreline spread in the streamwise direction more than those in the shear region; (ii) the spreadings in the streamwise and transverse directions differ at the centreline, while being similar in the shear region, (iii) spreading in cross-sections takes place in a similar way along radial and tangential directions.

Experimental Measurements of the Spatial Distribution of 3-D Lagrangian Dispersion Rates.

The mixing phenomenon in fluid flow fields has two major components : the advective mixing process driven by the large scale fluid motions and the diffusive mixing process driven by small scale molecular interactions. It is well accepted and amply demonstrated that, in many flows of interest to engineering the overall mixing can be strongly enhanced by an efficient advective mixing, i.e.by bringing the different species involved in close proximity to each other for molecular diffusion to take place over a short period of time. Over the past several years, we have developed an experimental 3-D fluid measurement technique, namely 3-D particle tracking velocimetry(PTV), in which the 3-D fluid velocities are determined by establishing(by photographic means)the 3-D trajectories of flow tracer particles. This paper describes a method for extracting 3-D Lagrangian dispersion rates from such 3-D PTV measurements. Specifically, a quantitative measure of the advective mixing is obtained by evaluating the normal dispersion rate of instantaneous nearby physical trajectories(in direction normal to the instantaneous local velocity vector). Then the magnitude of the root mean square(rms)of the instantaneous local dispersion rate is computed(in the ensemble averaged sense)over many realizations throughout the entire volume of the fluid being studied. This technique is applied to an in-cylinder flow field in an internal combustion(IC)engine in order to promote a better under-standing of mixing characteristics by comparing the spatial distribution of the normal dispersion to the 3-D flow patterns.

Lagrangian Particle Simulation of Tracer Dispersion in the Lee of a Schematic Two-Dimensional Hill

Abstract Spray, a 3D Lagrangian particle model for the simulation of complex flow dispersion, is presented. Its performance is tested against the Environmental Protection Agency wind tunnel concentration distribution of passive tracer released from elevated point sources, located in the lee region of a two-dimensional schematic hill, in a neutrally stratified boundary layer. Based on the measured values of the first two moments of the turbulent flow velocity, the mean fields are computed over a regular grid using a mass-consistent model, whereas the turbulence structure is simply interpolated. From these fields, trajectories of tracer particles are computed using a linear formulation of the Langevin equation, with a correlated, skewed forcing. The self-consistence test (well-mixed condition), aimed at maintaining an initially well-mixed particle distribution uniform in time, has shown satisfactory results in the region under study. The computed concentration field turns out to be in good agreement with th...

Trajectories of charged tracer particles around a charged sphere in a simple shear flow

The trajectories of electrically charged tracer particles travelling around a charged sphere subjected to a simple shear flow have been calculated. This is a limiting case of the relative trajectories of two unequal-sized spheres when the radius ratio a1/a2 approaches zero. Until now these trajectories have been calculated by assuming the additivity of hydrodynamic and electrostatic forces, while neglecting the electroviscous coupling forces. These electroviscous forces are long range and can significantly alter the relative trajectories of spheres. When a1/a2 → 0, it is found that these trajectories depend on two parameters, α and β, which depend on the surface charge density of the tracer particle and the sphere. The relative trajectories of charged particles are qualitatively different from those of neutral particles. There exist six intervals of α-values for which the trajectories of the tracer particle show different features. Several new types of trajectory appear, besides the open and closed trajectories for neutral particles, which we refer to as uni- and bidirectional infinite length trajectories, uni- and bidirectional finite length trajectories, open returning trajectories, and prolate, oblate and circular closed trajectories. This richness of possible trajectories is the result of three electrokinetic phenomena, affecting particle motion: electro-osmotic slip, electrophoretic and diffusiophoretic motion.

Mixing of a Granular Material in a Bidimensional Rotating Drum

AbstractWe report experimental measurements on mixing properties in bidimensional rotating drum. Using an image processing device, we follow the trajectories of tracer particles in a monodisperse assembly of beads. Tracer particles with different size ratios exhibit a violent segregation effect: a smaller particle has a tendency to stay in the centre and a larger one will rather dwell on the edges. Furthermore, for a tracer of identical size, we evidence a specific dispersion property where the centre and the edges are competing attractors of the mixing dynamics.

Reconstruction of three-dimensional particle trajectories in flows through curved circular tubes

An automated technique is described for reconstructing three-dimensional trajectories of tracer particles in curved circular ducts. Individual particles are tracked in real time by a rotating camera under computer control. A digital imaging system enables the computer to locate the particle, adjust the speed of rotation, and store position and calibration data. By viewing the tube from approximately orthogonal directions, three-dimensional information on the position of the particle is obtained. Its precise location is calculated by tracing rays from the camera to the interior of the tube. This technique yields detailed three-dimensional position and velocity data along a trajectory.

Simultaneous Measurement of Bubbles and Solid Particles in a Gas-Solid Fluidized Bed.

A simultaneous and continuous measuring method for the gas bubbles and solid particles is developed by introducing high-shutter speed video cameras with 1/2000s exposure time and an image analysis system. The image analysis system includes a color special-effects generator, with which the images from each of two video cameras can be superimposed, and a video disk recorder which can digitize and store 1200 images successively.The system was inspected by using a single-layer two-dimensional fluidized bed. Both the dilute portion around gas bubbles and the trajectories of tracer particles are clearly observed. As a result, it can be said that the video system introduced to this work is quite effective for both macroscopic and microscopic investigation of the gas-solid fluidized bed.

Comment on 'Flow visualisation with stroboscopic illumination to provide an unambiguous record of direction'

A recent article (Hampshire 1984) describes a stroboscopic illumination technique for determining the direction of motion of tracer particles illuminated by a beam of light. The required temporal marking of particle trajectories is achieved by varying the period of illumination using a mechanical chopper which has been modified to give flashes of unequal duration separated by a short extinction. A previously published paper (Popovich and Weinberg 1983) includes a description of an identical technique developed for use in flow visualisation studies of turbulent buoyant flows produced by room fires. The basic principle of the modified stroboscopic illumination technique is that apertures are cut out of a circular chopper blade in such a way that their areas are in some fixed ratio, which gives rise to illumination which is modulated so that the periods of illumination are in the same ratio. An automatic image-analysis system based on a TV camera interfaced to a dedicated microprocessor has been used for the analysis of tracer particle trajectories in turbulent fire plumes, recorded on 16 mm cine film using the modified stroboscopic illumination technique (Popovich, 1982).