Method Of Concentration Moments Research Articles

Effects of wind on transient dispersion of active particles in a free-surface wetland flow

Mechanisms of active-particle transport in a wetland flow under wind are significant for understanding various biological and ecological processes associated with wetlands. A transport model has been formulated to characterize the transient dispersion of active particles in a typical free-surface wetland flow exposed to wind. The transient evolution of total quantity, centred moving velocity, longitudinal dispersivity, skewness, and kurtosis of the active-particle cloud are analysed, using concentration moment method. It is found that the migration of active-particle cloud can eventually reach a stable status under the combined action of wind, hydrodynamics, and cell motility. The strong reverse wind can result in the negative centred moving velocity and longitudinal dispersivity along each streamline. The strong wind can considerably change the skewness and kurtosis of cell concentration distribution. The analytical solution of depth-averaged and two-dimensional concentration of active particles are derived based on Chatwin’s long-time expansion method, showing a distinct deviation from Gaussian distribution during the initial stage, which is in good agreement with the concentration moment analysis.

Transport model of active particles in a tidal wetland flow

A concentration transport model has been formulated to characterize the transport of a mass of self-propelling active particles in an unsteady wetland flow at the phase-averaged scale. The vertical distribution and motion of mass center are rigorously derived for an active-particle cloud in the tidal wetland flow by use of the concentration moment method, while the longitudinal dispersivity is calculated by means of finite difference method. It is found that the distribution of total quantity of active particles in each streamline in the tidal wetland flow is identical to that in the fully developed wetland flow. The velocity and longitudinal dispersivity of mass center of the active-particle cloud eventually reach a stable oscillatory status for the case of tidal wetland flow rather than an asymptotic constant velocity which occurs in the steady wetland flow. The effects of dimensionless parameters on the amplitude, phase and period of the velocity of mass center and longitudinal dispersivity are discussed in detail for the active-particle cloud in the tidal wetland flow.

Analyzing solute transport in modeled wetland flows under surface wind and bed absorption conditions

Both surface wind and bed absorption can greatly impact the pollutant dispersion processes, however, the two factors are not jointly considered in existing models due to the intricacy of analyzing the simultaneous influence of surface wind shear and bottom bed absorption. In this work, a multi-scale approach is employed to formulate the environmental dispersion process with both factors considered jointly within wetland flows. By adopting this method, the effect of wind and bed absorption on vertical concentration distribution and longitudinal concentration distribution are discussed. To visualize the results, the flow velocity contours under different wind conditions (i.e. upstream wind and downstream wind), the pollution concentration distributions versus time and spatial position, and the spatial contaminant cloud contours are presented. Moreover, numerical experiments of the interested scenarios are presented to verify the accuracy of the analytical method. The comparison shows that the results of the proposed multi-scale method agrees well with the dataset generated numerically. Finally, the multi-scale method in this work is compared to the Aris? concentration moment method in the literature by setting the bed absorption rate to zero. The comparison shows that the present model can reduce the substantial integration processes and derive accurate spatial concentration profiles.

On the bicomponent contaminant transport in wetland flow with reactions

Contaminant transport in the wetland flow is subject to various reactions. Most researches on this topic have limited to the single component contaminant, relatively little has concerned the effect of competitive reaction on the transport of coupled bicomponent contaminant. This work analytically studies the bicomponent contaminant transport in the free surface wetland flow under the combined conditions of reversible competitive transfer, irreversible degradation and bed absorption, by solving the constant-coefficient second-order linear system of parabolic type and the method of concentration moments. Up to the fourth order concentration moment in the previous work (Wang and Chen, 2017a) is applied to support the fourth order expansion of Hermite polynomials to rigorously derive the solutions with high accuracy. The results demonstrate that the masses of binary components decay at different rate, and the vertical concentration distributions of binary components are tremendously non-uniform in the asymptotic dispersion stage. Three types of reactions in addition to the hydraulic dispersion exert separate control on the concentration distributions. It suggests that the peak concentration rather than the mean should be based for strict environmental implements.

Transient dispersion of an initial point pollutant concentration in wetland flows.

Initial point pollutant transport in wetland flows acts as a typical form in various applications. Accurate determination of the basic characteristics for dispersion fundamentally supports the predication of pollutant transport especially in the transient stage. This work analytically studies the transient dispersion of an initial point pollutant concentration in wetland flows through the phase average technique and the method of concentration moments. The characteristics including the vertical mass distribution, longitudinal displacement, dispersivity, skewness, and kurtosis corresponding to the zeroth- to fourth-order concentration moments are accurately illustrated in their full-time dependance for the first time. Present results show to be distinctive from those of initial uniform concentration in the transient dispersion process. The temporal evolutions of the five characteristics are all dependent of the initial point release position. The results further demonstrate that the timescale 3.0 as measured by the skewness, rather than 0.5 as measured by the dispersivity, should be characterized as the transient period for the dispersion of initial point concentration in comparison with that of initial uniform. Furthermore, the longitudinally distributed mean concentration is as well analyzed through the Hermite polynomials. The results imply that strict criteria for environmental and ecological implications should be established by further accounting for the five characteristics.

Indicators for evaluating uncertainties of solute concentration prediction in wetland flows

Abstract Solute transport in wetland flows is an important process to understand for agricultural practices, providing essential information to assist in decision making and regulation for both cases of agricultural water contamination and agricultural pollution. Recent study has suggested that in the application of the one-dimensional Taylor dispersion model for solute concentration distribution in wetlands, uncertainties can arise even when contaminants are initially released at the same streamwise location. In this paper, we present an analytical study of instantaneous point-source release of solute in a wetland flow dominated by bank wall effects. The movement of the solute centroid and the long-term asymptotic Taylor dispersion coefficient are determined by the Aris concentration moment method, which are required for quantifying the streamwise distribution of the mean concentration and the transverse concentration distribution profiles. It is shown that the additional longitudinal displacement serves as a key indicator revealing the uncertainties of concentration distribution caused by different lateral positions of the contaminant release. More indicators including the lateral concentration distribution and the relative difference of the concentration are analyzed in detail. With given physical parameters, simple applications are provided to illustrate the streamwise transport of the solute plume centroid and the evolution of the additional longitudinal displacement affected by the bank wall effects.

Concentration distribution of environmental dispersion in a wetland flow: Extended solution

Hydrological processes of contaminant transport in wetland flows are characterized by environmental dispersion. For concentration distribution of scalar dispersion, the preliminary estimate of vertical distribution (Wu et al., 2015) on small time scale is rigorously extended to account for high order effects as skewness and kurtosis. Based on a combination of Aris’ method of concentration moments and Gill’s generalized dispersion model, up to fourth order terms of vertical distribution functions and full-time two-dimensional concentration distribution are derived. The increments of damping factor, mainly representing the density of vegetation in wetlands, is shown to strengthen the mean distribution asymmetry and reduce the longitudinal dispersivity (with respect to skewness and kurtosis), and to increase the non-uniformity of vertical distribution. Therefore, dense vegetation corresponding to a large damping factor will postpone the time when Taylor’s dispersion model holds, and the first order approximation becomes rough on small time scale, demanding higher order modifications. The results provide more detailed and more accurate analytical solution for the typical environmental dispersion in wetland flows.

Indicators for environmental dispersion in a two-layer wetland flow with effect of wind

For typical two-layer wetland flows where wind occurs frequently, indicators for flow and contaminant dispersion with wind effect are considered in this paper. Based on the general formulation for porous media flows, the velocity distribution of a fully developed flow through a wetland is illustrated, with that for a single-layer wetland flow under wind recovered as a special case. And the contrary wind of high speed might result in the appearance of an inverse-flow layer. For the more complex case of a two-layer wetland flow under wind, the more simplified method of asymptotic analysis is adopted instead of Aris's method of concentration moments to determine the dispersivity in terms of the longitudinal evolution of the depth-averaged concentration. And it is illustrated with the effect of dimensionless parameters. The velocity profile and environmental dispersivity for the two-layer wetland flow in the absence of wind can be included as a limiting case. Both the direction and the strength of the wind can affect the environmental dispersivity considerably under long-term evolution. Analytical solutions for the mean concentration and the region influenced by the contaminant cloud are derived. And for the instantaneous emission of typical contaminant constituents, the related indicators for water quality assessment, such as the critical length and duration of the influenced region are illustrated. The results show that the critical length and duration of the concentration cloud can change considerably because of the wind and vegetation heterogeneity.

Basic characteristics of Taylor dispersion in a laminar tube flow with wall absorption: Exchange rate, advection velocity, dispersivity, skewness and kurtosis in their full time dependance

Taylor dispersion with wall absorption in a laminar tube flow is analytically studied in this work by Aris’s method of concentration moments. The difficulty associated with the conventional approach initiated by Sankarasubramanian and Gill is completely avoided. All the basic characteristic properties governing the dispersion process, including the exchange rate, advection velocity, dispersivity, skewness and kurtosis, are analytically determined in their full time dependance for the first time. Detailed parametrical analysis is performed for the properties. Mean concentration distribution is concretely characterized.

Axial diffusion effect on concentration dispersion

This work presents an analytical attempt to illustrate the effect of axial diffusion on concentration dispersion in packed tube flows. The concentration moment method is applied to derive the analytical solution for time-dependent Taylor dispersion coefficient. With the increase in the packing parameter α as a damping factor, the concentration dispersion is characterized by three modes: (I) the shear-induced dispersion dominated mode, with dispersion coefficient determined by α; (II) the transition mode; and (III) the diffusion dominated mode, with dispersion coefficient determined by Péclet number Pe. In contrast to the usually negligible effects of axial diffusion for dispersion in pure fluid flows even at a relatively small Pe, axial diffusion with typical α plays an important role for packed media flows even with large Pe.

Two-scale analysis for environmental dispersion in a two-layer wetland

Studies on environmental dispersion are essential for applications as water management. The two-scale perturbation analysis is applied in this paper to deduce the environmental dispersion model for the typical case of contaminant transport in two-layer wetland flows. The analysis follows the established theoretical framework on the basis of phase average and the concept of Taylor dispersion. By the obtained flow velocity distribution for the two-layer flow, the analytical expression for the environmental dispersivity is deduced and shown to be consistent with previous results by the concentration moment method, while with much simplifications on the expression for ignoring the less concerned time-dependent stage of the dispersivity.

The asymptotic time variation of Taylor dispersivity for scalar transport in a two-zone packed tube

For transient scalar transport in a tube packed with two-zone heterogeneous porous media, presented in this paper is an extension of the existing study to determine the steady dispersivity for the long term process of Taylor dispersion (Chen and Wu, 2012). For the initial phase of Taylor dispersion with essential implications to various applications, this work aims to illustrate the asymptotic time variation of the dispersivity, by means of Aris’s method of concentration moments originally devised for the analysis of dispersion in a flow through an empty tube. The analytical solution of the dispersivity covering both the initial and long term phases is rigorously derived to reflect the asymptotic structure of the dispersivity as characterized by relevant dimensionless parameters which represent the relative magnitude of properties and the combined influences of two zones.

Environmental dispersion in a tidal wetland with sorption by vegetation

Understanding of the solute transport mechanism under the effect of sorption by vegetation in tidal wetland gains its significance for environmental and ecological management. Presented in this paper is a theoretical analysis of effective environmental dispersion in a depth-dominated tidal wetland. Based on the transport in porous media, a linear sorption isotherm model is adopted to account for the sorption by vegetation, and two models for momentum and concentration transport in wetlands are given, respectively. The velocity of flow forced by oscillating pressure is derived, and the effect of dimensionless parameters on velocity pulsation is analyzed. The velocity direction may reverse in the case of pulsation amplitude larger than the mean velocity. Using Aris’s method of concentration moments, we investigate the effective environmental dispersivity and concentration distribution. The effective environmental dispersivity increases over time at the initial stage to attain a steady oscillating status, the growth rate of which depends on the distribution coefficient KD. The variations of concentration distribution with typical dimensionless parameters are determined, which turn out to be consistent with those of dispersivity. The sorption by vegetation leads to lowered concentration and delayed contaminant cloud, contributing to the dispersion.

Indicators for contaminant transport in wetlands

Wetlands provide significant ecological services for urban regions in terms of water supply, wastewater treatment, flood storage, drought resistance, etc. For wetland flows, it is crucial to understand the process of contaminant transport as it provides scientific support for applications associated with various urban services. Two indicators respectively as the critical length and duration are frequently adopted for risk assessment of incidental release of toxic or contaminant cloud. This paper presents a review on recent progresses in the analytical study of contaminant transport in wetland flows by Taylor dispersion at the phase-average scale. The method of concentration moments is introduced. Analytical procedures for determining the key quantity of Taylor dispersivity are given for typical wetlands with free water surface, respectively as the steady flow wetlands, tidal flow wetlands, and the two-layer flow wetlands. As an example of applications, critical length and duration of the contaminant cloud in the steady flow wetlands are analyzed based on the obtained Taylor dispersivity. Results show that in contrast to the temporary, localized influence of COD on water quality, the heavy metal Pb can give rise to more severe damage.

Contaminant transport in a two-zone wetland: Dispersion and ecological degradation

Understanding the fate of contaminant in wetland flows is essential in applications such as ecological risk assessment and environmental hydraulic design. Presented in this paper is an analytical study on the dispersion of contaminant in a two-zone wetland, with the effect of ecological degradation taken into consideration. Environmental dispersion is discussed separately via an exponential transformation for the general formulation of contaminant transport. Taylor’s classical analysis for solute dispersion in a long and thin tube flow is rigorously generalized for the dispersion of the lateral mean contaminant concentration in the longitudinal direction. A method of asymptotic analysis is adopted instead of the concentration moment method in order to simplify the process of deduction and the expression of the analytical solution. Gill’s method of mean concentration expansion is applied to model the concentration deviation terms produced in an averaging operation. With the velocity profile obtained previously, the enhancement of the environmental dispersivity under long time evolution is determined and shown to be consistent with that obtained by the method of concentration moment. Analytical solutions for the evolution of the contaminant concentration and the influenced region of the contaminant cloud are obtained by combining both the hydraulic and the ecological effects. For typical pollutant as the heavy metal Hg, the evolution of contaminant cloud is illustrated by critical length and duration in an application with concentration beyond some given environmental standard level. Results show that for wetland flows with a two-zone structure, the influenced region is reduced evidently while the duration of the contaminant cloud remains nearly unchanged, compared with that for the single-zone wetland flow.

Environmental dispersion in a tidal flow through a depth-dominated wetland

Presented in this paper is a theoretical analysis for longitudinal evolution of mean concentration of an environmental emission into a tidal wetland flow. The velocity distribution of the periodic flow through the wetland is derived, with that for a fully developed steady wetland flow included as a special case. The zero-th, first and the growth of the second order moments of the concentration are rigorously obtained by applying Aris’s method of concentration moments to derive the environmental dispersivity. The necessary time needed for the environmental dispersivity to attain a steady oscillating status is analyzed. The effects of some characteristic parameters, especially one representing the impact of vegetation in the wetland on both velocity profile and environmental dispersivity, and another identifying the effect of flow oscillation on the environmental dispersivity, are illustrated in detail. To reflect the dispersion enhancement by the flow oscillation, a typical example is given to characterize the critical length and duration of the contaminant cloud.

Effect of wind on contaminant dispersion in a wetland flow dominated by free-surface effect

Flow and environmental dispersion are essential to ecological risk assessment and ecological restoration associated with wetlands. Presented in this paper is an analytical study for effect of wind on contaminant dispersion in a wetland flow dominated by free-surface effect. Based on the momentum transport equation generally adopted for wetland flow, the solution for velocity distribution in a free-surface wetland flow under wind is rigorously derived, with a limiting case covering the known solution of flow in the absence of wind. The thickness of inverse flow layer, maximum velocity, and its location are obtained. By use of Aris's method of concentration moments, the expression for environmental dispersivity in the free-surface wetland flow under wind is derived, and that in the absence of wind can be included as a special case. The direction and intensity of wind can influence greatly the environmental dispersivity at a stable stage. The time scale for environmental dispersivity to approach the stable stage is independent of wind. Analytical solution of mean concentration is derived and illustrated for a pulsed contaminant emission into a wetland flow dominated by free-surface effect under wind. For typical pollutant Pb, the length and duration of influence region with contaminant concentration beyond given standard level of water quality are illustrated.

Flow distribution and environmental dispersivity in a tidal wetland channel of rectangular cross-section

Presented in this paper is a theoretical analysis on flow distribution and environmental dispersivity for a tidal wetland channel of rectangular cross-section. The analytical solution of velocity distribution for the tidal wetland flow is obtained and illustrated with a limiting case covering the known solution for a steady wetland flow. By use of Aris’s method of concentration moments, the environmental dispersivity for a pulsed contaminant emission into the tidal wetland flow is rigorously derived and characterized in terms of dimensionless parameters. The solution is shown to be a generalization of the environmental dispersivity for the corresponding steady wetland flow, taking into account the combined action of periodic oscillation and cross-sectional variation of superficial flow as well as the difference between superficial mass dispersivities in the vertical and lateral directions. For a long time evolution of the contaminant cloud, the environmental dispersivity may approach a stable stage of oscillation with a period equal to the period of the superficial flow. The evolution of environmental dispersivity at the initial stage for the tidal wetland flow is shown not monotonous as it does in the case of the steady wetland flow. It is also found that the period of superficial flow has no impact on the necessary time for the environmental dispersivity to attain the stable stage.

Environmental dispersion in a two-layer wetland: Analytical solution by method of concentration moments

For the most typical two-layer wetland with free-water-surface-effect, analytically explored in this paper is the environmental dispersion in terms of the longitudinal evolution of the depth-averaged concentration. Taylor’s classical analysis for soluble matter dispersion in a single phase fluid flow is rigorously generalized for the two-layer case of a wetland flow to develop the dispersion model. The velocity profile of the fully developed flow through the wetland is derived, with that for the single-layer wetland flow included as a special case. Aris’s method of concentration moments is applied to determine the dispersivity with an asymptotic time variation. For typical contaminant constituents of chemical oxygen demand, biochemical oxygen demand, total phosphorus, total nitrogen and heavy metal, the process of dispersion is shown characteristic of a hierarchical structure in the critical length and duration of the contaminant cloud.

Environmental dispersivity in free-water-surface-effect dominated wetland: multi-scale analysis

Distinct from the case with width-dominated shallow wetland flows, the longitudinal evolution of contaminant concentration in the most-typical pattern of wetland as dominated by free-water-surface-effect is characterized by a multi-scale analysis in the present study. An environmental dispersion model for the evolution of the mean concentration is deduced as an extension of Taylor’s classical formulation by Mei’s multi-scale analysis. Corresponding environmental dispersivity is found identical to that determined by the method of concentration moments.