Effective Longitudinal Dispersion Coefficient Research Articles

Upstream swimming and Taylor dispersion of active Brownian particles

Locomotion of self-propelled particles such as motile bacteria or phoretic swimmers often takes place in the presence of applied flows and confining boundaries. Interactions of these active swimmers with the flow environment are important for the understanding of many biological processes, including infection by motile bacteria and the formation of biofilms. Recent experimental and theoretical works have shown that active particles in a Poiseuille flow exhibit interesting dynamics including accumulation at the wall and upstream swimming. Compared to the well-studied Taylor dispersion of passive Brownian particles, a theoretical understanding of the transport of active Brownian particles (ABPs) in a pressure-driven flow is relatively less developed. In this paper, employing a small wave-number expansion of the Smoluchowski equation describing the particle distribution, we explicitly derive an effective advection-diffusion equation for the cross-sectional average of the particle number density in Fourier space. We characterize the average drift (specifically upstream swimming) and effective longitudinal dispersion coefficient of active particles in relation to the flow speed, the intrinsic swimming speed of the active particles, their Brownian diffusion, and the degree of confinement. In contrast to passive Brownian particles, both the average drift and the longitudinal dispersivity of ABPs exhibit a nonmonotonic variation as a function of the flow speed. In particular, the dispersion of ABPs includes the classical shear-enhanced (Taylor) dispersion and an active contribution called the swim diffusivity. In the absence of translational diffusion, the classical Taylor dispersion is absent and we observe a giant longitudinal dispersion in the strong flow limit. Our continuum theory is corroborated by a direct Brownian dynamics simulation of the Langevin equations governing the motion of each ABP.

Random walk particle tracking simulation on scalar diffusion with irreversible first-order absorption boundaries.

Scalar transport in an open channel with irreversible first-order absorption boundaries is investigated through random walk particle tracking method (RWPT). We provide the pre-asymptotic behavior of scalar transport as well as a comparison with existing asymptotic formulations. The RWPT method is based on the development of the probability that a particle is absorbed at the boundary. The random walk model is applied to simulate three main parameters of the scalar transport in the laminar flow with sorption boundaries. The three parameters are (1) the attenuation coefficient reflecting the depletion rate of the total mass; (2) the effective velocity coefficient representing the effective advection of the total mass; and (3) the effective longitudinal dispersion coefficient expressing the effective spreading rate of the solute plume. When the Peclet number is large, numerical results agree well with previous theoretical solutions. Results show that (1) the three coefficients reach their asymptotic values when the dimensionless time τ (= Dt/H2) is about 0.5, where t is the time, H is the water depth, and D is the molecular diffusion coefficient and (2) when the Damkohler number reaches a value around 100, the boundary can be treated as a totally absorptive boundary.

Pore-Network Simulation of Unstable Miscible Displacements in Porous Media

We develop a comprehensive pore-network model for simulation of miscible displacements in laboratory-scale porous media. The goal is to study the effect of pore-scale heterogeneity, and in particular pore-size distribution and pore connectivity, as well as mobility ratio M on miscible displacements, and compute the effective longitudinal dispersion coefficient as a function of M. At the pore level the phenomenon is described by the convective-diffusion equation with a Taylor–Aris dispersion coefficient. Extensive simulations indicate the strong effect of the connectivity and broadness of the pore-size distribution on the efficiency of the displacement process. Moreover, the model enables us to compute the effective dispersion coefficient as a function of the mobility ratio of the two fluids that, to our knowledge, has not been calculated before. Simple generalization of the model will make it possible to study more complex related phenomena, and in particular injection of $$\mathrm{CO}_2$$ into a pore space that contains saline water.

Spatio‐temporal distribution, along‐channel transport, and post‐riverflood recovery of salinity in the Guadalquivir estuary (SW Spain)

This paper presents an experimental analysis of the salinity distribution, the salt balance, and the variation of the saline intrusion in comparison to the freshwater discharge in the Guadalquivir estuary, which is a mesotidal system regulated and normally subjected to extremely low river flows. In such low‐flow conditions, it is positive, well‐mixed, and tidally dominated. The estuary is also characterized by a nonstationary, effective longitudinal dispersion coefficient, whose probability density becomes increasingly narrower and whose mean value is higher further upstream. The tidal‐averaged salt flux is controlled by the following mechanisms (in order of importance): the nontidal transport, the Stokes transport, and the tidal pumping induced by the covariance between the current and salinity. These three factors account for more than 98% of the flux variation. In high river‐flow conditions, the subtidal response and recovery of the estuary to changes in the river flow is analyzed. The increase in the tidal‐averaged salinity during the first 2 weeks of the post‐riverflood recovery in the middle and upper sections of the estuary is found to be linear in time. During that time, the celerity of the salt intrusion front was 4cm/s. The 2 psu isohaline salt intrusion X2 exhibits a complex dependence on the river flow Qd, including the effects of human interventions in the estuary. Three regimes are identified for the intrusion: X2=57.0± 2.1km for discharges of less than 20m 3/s, X2 proportional to between 20 and 1000m 3/s, and X2proportional to for larger discharges.

Spatial variation of the longitudinal dispersion coefficient in an estuary

Abstract. The effective longitudinal dispersion is a primary tool for determining property distributions in estuaries. Most previous studies have examined the longitudinal dispersion coefficient for the average tidal condition. However, information on spatial and temporal variations of this coefficient at low and high tides is scarce. Three years of hydrographic data taken at low and high tide along the main axis of the Sumjin River Estuary (SRE), Korea are used to estimate the spatial and temporal variation of the effective longitudinal dispersion coefficient. The range of the dispersion coefficient is rather broad at high water slack (HWS) and narrower at low water slack (LWS) because of the different tidal amplitudes. The spatially varying dispersion coefficient has maximal values (>300 m2 s−1) near the mouth at high water and decreases gradually upstream, with fluctuations. The temporally varying dispersion coefficient appears to be positively correlated with river discharges at both low and high tide. The dispersion varies with the square root of river discharges at HWS and LWS. The dispersive salt fluxes increases with increasing river discharges and decreases with decreasing river discharges at HWS and LWS. Estimation of the numerical values of the effective longitudinal dispersion coefficient in the SRE can be useful for better understanding of the distributions of other tracers in the SRE as well as for developing and testing hypotheses about various mixing mechanisms.

Stochastic modeling of reactive transport in wetlands

This study describes the development of a general model for reaction in and performance of spatially heterogeneous bioreactors such as treatment wetlands. The modeled domain possesses local-scale velocities, reaction rates and transverse dispersion coefficients that are functions of an underlying heterogeneity variate representing one or more controlling biophysical attributes, for example, reactive surface area (submerged plant) density. Reaction rate coefficients are treated as related to local velocities in an inverse square fashion via their mutual dependence upon the variate. The study focuses on the solution for the steady-state case with constant inlet concentration. Results compare well with exact solutions developed for laterally-bounded systems in which the heterogeneity is represented explicitly. Employing the bicontinuum analogue of a second-order model, an expression for an effective longitudinal dispersion coefficient as a function of travel distance is developed using the method of moments. The result provides insights into the behavior of concentration as transverse mixing drives the system asymptotically toward Fickian longitudinal dispersion. The model may represent an improvement over other approaches for characterizing treatment wetland performance because it accounts for evolving shear flow dispersion, and because parameters are few in number, physically based, and invariant with mean velocity.

Estimating effective longitudinal dispersion in the Chesapeake Bay

An analysis of Environmental Protection Agency's Chesapeake Bay Program hydrographic dataset shows that the bay responds coherently to variability in freshwater flux. Mean salinity and salinity stratification both respond to variability in freshwater flux on time scales of roughly 90 days. Stratification is also influenced by local wind forcing but on much shorter (4–5 day) time scales. The volume of available data allows the effective longitudinal dispersion coefficient to be estimated as a function of either time or space. Values for this dispersion coefficient vary between 200 and 1000 m 2 s −1, with mean values around 650 m 2 s −1. The spatially dependent structure has a maximum roughly 75 km from the head of the estuary, and decreases gradually towards the mouth. The temporally varying effective dispersion varies spatially as the inverse of the estuarine cross-section, and temporally as the cube root of the freshwater flux, and is at least qualitatively consistent with models of estuarine circulation and results of previous field studies. Estimates of the numerical values of the dispersion are useful for better understanding distributions of other tracers within the bay, as well as providing another metric against which numerical models should be measured.

Solute transport in variable‐aperture fractures: An investigation of the relative importance of Taylor dispersion and macrodispersion

Dispersion of solutes in a variable aperture fracture results from a combination of molecular diffusion and velocity variations in both the plane of the fracture (macrodispersion) and across the fracture aperture (Taylor dispersion). We use a combination of physical experiments and computational simulations to test a theoretical model in which the effective longitudinal dispersion coefficient DL is expressed as a sum of the contributions of these three dispersive mechanisms. The combined influence of Taylor dispersion and macrodispersion results in a nonlinear dependence of DL on the Peclet number (Pe = V〈b〉/Dm, where V is the mean solute velocity, 〈b〉 is the mean aperture, and Dm is the molecular diffusion coefficient). Three distinct dispersion regimes become evident: For small Pe (Pe ≪1), molecular diffusion dominates resulting in DL ∝ Pe0; for intermediate Pe, macrodispersion dominates (DL ∝ Pe); and for large Pe, Taylor dispersion dominates (DL ∝ Pe2). The Pe range corresponding to these different regimes is controlled by the statistics of the aperture field. In particular, the upper limit of Pe corresponding to the macrodispersion regime increases as the macrodispersivity increases. Physical experiments in an analog, rough‐walled fracture confirm the nonlinear Pe dependence of DL predicted by the theoretical model. However, the theoretical model underestimates the magnitude of DL. Computational simulations, using a particle‐tracking algorithm that incorporates all three dispersive mechanisms, agree very closely with the theoretical model predictions. The close agreement between the theoretical model and computational simulations is largely because, in both cases, the Reynolds equation describes the flow field in the fracture. The discrepancy between theoretical model predictions and DL estimated from the physical experiments appears to be largely due to deviations from the local cubic law assumed by the Reynolds equation.

The influence of surface sorption on dispersion in parallel plate fractures

Solute transport in a parallel plate fracture is simulated using a random walk model which accounts explicitly for sorption onto the fracture walls. With the hypothesis that solute must move into the vicinity of the fracture wall in order to participate in the sorption process, three implications follow. (1) In comparison to a nonreactive solute, a sorbing solute requires a greater entrance length along the fracture before transverse homogenization is established. The increase in this entrance length is proportional to the surface retardation factor. (2) At short transport distances, surface sorption leads to a high degree of non-uniform retardation. The transport distance required to establish uniform retardation of the entire solute mass varies greatly with geometric conditions and sorption strength. (3) Surface sorption results in enhanced longitudinal spreading of the solute mass in transport regimes which favor advective transport along the fracture relative to transverse diffusion across the fracture aperture. At distances greater than that required for transverse homogenization, an effective longitudinal dispersion coefficient can be defined that describes this enhanced dispersion for a wide range of fluid velocities. The magnitude of enhanced dispersion increases with sorption strength.

Stochastic Theory for Irregular Stream Modeling. II: Solute Transport

A stochastic theory is developed for longitudinal dispersion in natural streams. Irregular variations in river width and bed elevation are conveniently represented as one-dimensional random fields. Longitudinal solute migration is described by a one-dimensional stochastic solute transport equation. When boundary variations are small and statistically homogeneous, the stochastic transport equation is solved in closed-form using a stochastic spectral technique. The results show that large scale longitudinal transport can be represented as a gradient dispersion process described by an effective longitudinal dispersion coefficient. The effective coefficient reflects longitudinal mixing due to flow variation both within the river cross section and along the flow and can be considerably greater than that of corresponding uniform channels. The discrepancy between uniform channels and natural rivers increases as the variances of river width and bed elevation increase, especially when the mean flow Froude number is high.

Dispersion of a passive solute in non-ergodic transport by steady velocity fields in heterogeneous formations

An inert solute is convected by a steady random velocity field, which is associated with flow through a heterogeneous porous formation. The log conductivity and the velocity are stationary random space functions. The log conductivity Y is assumed to be normal, with an isotropic two-point correlation of variance σY2 and of finite integral scale I. The solute cloud is of a finite input zone of lengthscale l. The transport is characterized with the aid of the spatial moments of the solute body. The effective dispersion coefficient is defined as half of the rate of change with time of the second spatial moment with respect to the centroid. Under the ergodic hypothesis, which is bound to be satisfied for l/I [Gt ] 1, the centroid moves with the mean velocity U and the longitudinal dispersion coefficient [dscr ]L tends to its constant, Fickian, limit. Under a Lagrangian first-order analysis in σY2 it has been found that [dscr ]L = σY2UI.This study addresses the computation of the effective longitudinal dispersion coefficient for a finite input zone, for which ergodic conditions may not be satisfied. In this case the centroid trajectory and the second spatial moments are random variables. In line with a previous work (Dagan 1990) the effective dispersion coefficient DL is defined as half the rate of change of the expected value of the second spatial moment for large transport time. The aim of the study is to derive DL and its dependence upon l/I and in particular to determine the conditions under which it tends to the ergodic limit [dscr ]L. The computation is carried out separately for a thin body aligned with the mean flow and one transverse to it. In the first case it is found that DL is equal to zero, i.e. the streamlined body does not disperse in the mean. This result is explained by the correlation between the trajectories of the leading and trailing edges, respectively, once the latter reaches the position of the first. The relatively modest increase of the mean second spatial moment is effectively computed. In the case of a thin body initially transverse to the mean flow, DL may reach the ergodic limit [dscr ]L for a ratio l/I of the order 102. For smaller values, DL is found to be bounded from above, and its maximum depends on l but not on I. The uncertainty caused by the randomness of the velocity field is manifested in the trajectory of the centroid rather than in the effective dispersion.

Characterization of a Malaysian Mangrove Estuary

The Sungai Merbok estuary, in wet tropical Peninsular Malaysia, borders the Straits of Malacca. Tide, current, and salinity data are used to describe the salient hydrographic features of the mangrove-fringed system. The Sungai Merbok estuary is characterized by a 1.7 m semidiurnal tide with a 0.16 form number, peak currents of 1.3 m s−1, and mean freshwater discharge of 20 m3 s−1. The system is classified as 2a/2b estuary (Hansen and Rattray 1966) or 1a/1b during periods of low runoff. Gravitational circulation is highly variable (but coincides with the neap stratification) and vertical stratification varies from 10−2 to 1. The estuary displays a pronounced fortnightly neap-spring stratification-destratification cycle. The effective longitudinal dispersion coefficient is approximately 100 m2 s−1.

Modulated mixing and frontogenesis in shallow seas and estuaries

This paper describes some laboratory experiments on the circulation produced in a fluid containing a horizontal density gradient in the presence of environmental turbulence whose level varies periodically with time. The experiments are intended to model the effects of turbulence generated by tidal flow in a shallow sea or estuary. At times of high tidal flow the water column is vertically mixed and the baroclinically driven circulation is weak. At low turbulence levels the baroclinic circulation accelerates and the water column stratifies. The longitudinal flux of density is measured and an effective longitudinal dispersion coefficient K is calculated. The value of K increases with the horizontal density gradient and also with the period of the turbulence modulation. This latter increase is a result of the enhanced flux driven by the baroclinic circulation at low turbulence levels. The results of these experiments are compared with some recent measurements of episodic mixing events in Liverpool Bay, U.K., Spencer Gulf, South Australia, and the Columbia River Estuary, U.S.A.

Dispersion Coefficients for Gases Flowing in Consolidated Porous Media

Abstract The best currently available description of the longitudinal mixing properties of a porous medium is an equation of the formEquation 1 which relates the effective longitudinal dispersion coefficient Dl to the molecular diffusion coefficient D0, the electrical resistivity factor F, the porosity f and a Peclet number. If the parameters dps and m are determined for a porous medium of known porosity and electrical resistivity factor, then a dispersion coefficient may be estimated for a given flow rate and a given gas pair. A new method, featuring on-line gas analysis by thermal conductivity and on-line data reduction by analog computation, was developed and used to determine these mixing parameters for eight naturally occurring sandstones and two dolomite samples. The exponent m of the above equation was found to vary between 1.0 and 1.5. The characteristic length dp s in the above equation was found to vary between 0.25 and 1.9 cm, with an average value of 0.4 cm for sandstones. Measurements were made on two cores in which paraffin wax had been deposited by evaporation from a pentane solution. They indicated that the presence of an immobile phase such as connate water could increase the dispersion coefficients significantly. INTRODUCTION While the petroleum and chemical industries have studied the mixing of miscible liquids flowing in consolidated porous media and of miscible gases flowing in unconsolidated porous media, relatively little data have been presented to describe the mixing of gases flowing through consolidated porous media. Such data are of particular interest to the gas storage industry. For instance, the U.S. Bureau of Mines is storing large quantities of a rich helium-nitrogen gas in contact with a natural gas in a dolomite reservoir. Since the rich gas occupies only 15 percent of the total reservoir volume, it is essential that the extent of rich gas-natural gas mixing be predicted and understood as a function of rock properties, pressure and rate of movement. This investigation was concerned only with the determination of longitudinal dispersion coefficients. It is understood that a transverse dispersion coefficient, which characterizes mixing perpendicular to the direction of flow, may be an order of magnitude less than the coefficient characterizing mixing in the direction of bulk flow.5,19 It should also be recognized that the use of any dispersion coefficient is in itself a simplification. It is necessary to assume that mixing in a porous medium may be characterized by the equationEquation 2 for flow in a single direction. A number of authors1 have pursued the mixing problem, not in terms of the so-called "dispersion model" described by Eq. 1, but in terms of a "mixing cell model". This model supposes that a porous medium is constructed of a large number of small mixing chambers and that the concentration of the diffusing component within each mixing chamber is uniform. Fick's law (Eq. 1) assumes that there is no gross by-passing of one fluid by another, and that there are not stagnant pockets of gas in the system under consideration as discussed by Coats and Smith.8 These assumptions are not always valid for flow through porous media and it is important to recognize the limitations upon Eq. 1.

Gaseous dispersion in laminar flow through a circular tube

The dispersion of a pulse of ethylene injected into nitrogen, flowing in the laminar régime through straight and curved tubes, has been investigated at pressures of 1.0 and 4.4 atm. From the study of the concentration profiles with a thermal conductivity detector (katharometer) it is found that the experimental results for gas velocities between 1.00 and 16.00 cm/s agree well with the analytical solution to this problem for a straight tube given by Sir Geoffrey Taylor and extended by Aris. In particular, at low velocities, the effective dispersion coefficients tend to the molecular diffusivities. The presence of a bend slightly reduces the effective longitudinal dispersion coefficient and the introduction of constrictions enhances it. Data are also given on a number of other gas pairs. It is concluded that measurements of dispersion provide an accurate and simple way of studying diffusion in gas mixtures.