Analytical solution of two-dimensional advection–dispersion equation with spatio-temporal coefficients for point sources in an infinite medium using Green’s function method

In the present study analytical solutions of a two-dimensional advection–dispersion equation (ADE) with spatially and temporally dependent longitudinal and lateral components of the dispersion coefficient and velocity are obtained using Green’s Function Method (GFM). These solutions describe solute transport in infinite horizontal groundwater flow, assimilating the spatio-temporal dependence of transport properties, dependence of dispersion coefficient on velocity, and the particulate heterogeneity of the aquifer. The solution is obtained in the general form of temporal dependence and the source term, from which solutions for instantaneous and continuous point sources are derived. The spatial dependence of groundwater velocity is considered non-homogeneous linear, whereas the dispersion coefficient is considered proportional to the square of spatial dependence of velocity. An asymptotically increasing temporal function is considered to illustrate the proposed solutions. The solutions are validated with the existing solutions derived from the proposed solutions in three special cases. The effect of spatially/temporally dependent heterogeneity on the solute transport is also demonstrated. To use the GFM, the ADE with spatio-temporally dependent coefficients is reduced to a dispersion equation with constant coefficients in terms of new position variables introduced through properly developed coordinate transformation equations. Also, a new time variable is introduced through a known transformation.     

Analytical solutions of nonlinear and variable-parameter transport equations for verification of numerical solvers

All numerical codes developed to solve the advection–diffusion-reaction (ADR) equation need to be verified before they are moved to the operational phase. In this paper, we initially provide four new one-dimensional analytical solutions designed to help code verification; these solutions are able to handle the challenges of the scalar transport equation including nonlinearity and spatiotemporal variability of the velocity and dispersion coefficient, and of the source term. Then, we present a solution of Burgers’ equation in a novel setup. Proposed solutions satisfy the continuity of mass for the ambient flow, which is a crucial factor for coupled hydrodynamics-transport solvers. By the end of the paper, we solve hypothetical test problems for each of the solutions numerically, and we use the derived analytical solutions for code verification. Finally, we provide assessments of results accuracy based on well-known model skill metrics.     

Secondary flow within a river bed and contaminant transport

We have developed a numerical method to simulate the transport of non-sorbing contaminants within the sediment layer of a stream and the leaching of these contaminants in the steam. Typical stream bottom surfaces are uneven with triangularly shaped undulation forms. The flow of the water above such triangular surfaces causes external pressure changes that result in a “pumping effect” and a secondary flow within the sediment. The latter causes a significant contaminant advection within the sediment layer. The flow field in the porous sediment layer is obtained by solving numerically Darcy’s equations. The unsteady mass transfer equation is solved by using a finite-difference method with an up-wind scheme. The effects of parameters, such as channel slope, hydraulic head and dispersion, are studied by quantitatively comparing the numerical results of the total mass flow rate from the contaminant source, the concentration front propagation, and the contaminant mass flow rate into the water column. The “pumping effect,” increases the flow in the vertical direction and, thus, enhances the vertical advective mass transport of the contaminant. This bedform-shape induced flow is largely responsible for the mass transfer of contaminants into the water column. The numerical results also show that the mechanical dispersion inside the sediment bed will significantly increase the contaminant mass flow rate from the source.     

An ecological risk assessment model for a pulsed contaminant emission into a wetland channel flow

As a continuation of the modelling on ecological degradation and hydraulic dispersion of pollutant emission into an idealized two-dimensional free-surface wetland flow (Zeng, L., Chen, G.Q., 2009b. Ecological degradation and hydraulic dispersion of contaminant in wetland. Ecol. Model., doi:10.1016/j.ecolmodel.2009.10.024), an ecological risk assessment model for the typical case of a pulsed contaminant emission into a realistic three-dimensional wetland channel flow is presented in this paper for the fate of cross-sectional mean concentration under environmental dispersion. An environmental dispersion model for the mean concentration is devised as an extension of Taylor’s classical analysis on dispersion in fluid flows. The velocity distribution and the environmental dispersivity in the fully developed steady flow through the wetland is found and illustrated with limiting cases covering various known solutions for the porous media flow between parallel plates, flow in a shallow wetland, sweeping flow in a densely vegetated wetland, and single phase flow in a channel. Obtained by Aris’s method of moments, the environmental dispersivity is shown characterized with multi-scale asymptotic time variations with stem dominated stage, transitional stage, and width-depth-stem dominated stage. Based on the solution for the evolution of contaminant cloud in the wetland channel flow, critical length and duration of the contaminant cloud with concentration beyond given environmental standard level are concretely illustrated for typical pollutant constituents in wastewater emission. Under the same emission intensity and environmental standard, the duration of contaminant cloud in the wetland channel is revealed shorter than that in a free surface wetland, due to the lateral effect.     

A new model for the CBL growth based on the turbulent kinetic energy equation

Starting from the evolution equation for the turbulent energy density spectrum (EDS), we develop a new model for the growth of the Convective boundary layer (CBL). We apply dimensional analysis to parameterize the unknown inertial transport and convective source term in the dynamic equation for the three-dimensional (3-D) spectrum and solve the 3-D EDS equation. The one-dimensional vertical spectrum is derived from the 3-D spectrum, employing a weight function. This allows us to select the magnitude of the vertical spectral component for the construction of the growing 3-D EDS. Furthermore, we employ the vertical component of the energy spectrum to calculate the eddy diffusivity (required in dispersion models). Currently there are no available experimental data to directly verify our EDS model.     

Comments on “Development of an empirical equation for the transverse dispersion coefficient in natural streams” by Tae Myoung Jeon,Kyong Oh Baek and Il Won Seo

In a recent paper published in this journal, Jeon et al. ((2007), Environ Fluid Mech 7(4): 317–329) have presented a new empirical equation for the transverse dispersion coefficient in natural streams that was developed based on the hydraulic and geometric parameters using a regression technique. A total of 32 data sets collected in 32 streams was used. Among them, 16 data sets were used for deriving the new equation, and the other 16 were used for verifying the equation. Starting from dimensional analysis the authors found that transverse dispersion coefficient depends on three parameters, such as sinuosity, aspect ratio and a friction term. The robust least square method was applied to estimate regression coefficients resulting in an equation which allows better prediction of transverse dispersion coefficient than previous literature equations. The discussers would like to highlight some points raised in the paper.     

Testing Accuracy of Finite Element and Random Walk Schemes in Prediction of Pollutant Dispersion in Coastal Waters

Distribution of pollutants in coastal waters is usually represented by depth averaged twodimensional convection-dispersion equation. Under very specific conditions this equation can be solved analytically. Although such a solution is restricted to simplified situations it provides a very useful case for testing the performance of various numerical solution techniques currently available for the simulation of convective-dispersion of pollutants in natural water systems. In this paper the analytical solution of the convective dispersion equation is used as a benchmark against which the accuracy of other techniques are assessed. These assessments are based on quantitative comparisons between the results of the solution of two-dimensional convection-dispersion equation by the deterministic finite element and stochastic random walk methods. Both Eulerian and Lagrangian frameworks are employed to obtain the finite element solution of the convection-dispersion problem. It has been shown that the Lagrange–Galerkin finite element scheme yields the most accurate results for the case under study. However, computational costs of the Lagrange–Galerkin method can be relatively high and under certain conditions it may be reasonable to use a less accurate but cost effective random walk scheme to make water quality management decisions.     

Solving unsteady problems in water pollution using Lie symmetries

This work proposes an analytic method to simulate the environmental damage caused by accidents with chemical substances in rivers and lakes. The two-dimensional transport equation is solved by rules based on the Lie algebra in situations in which the velocity field is considered locally constant. The exponential of the differential operator, which arises in the formal solution is locally approximated by the Taylor series. The initial condition is obtained by Lie symmetries. This method was by tested simulating scenarios that describe the blob's behavior through time and the results are compared with available data. Numerical results are shown.     

三氯生对青海弧菌Q67和人乳腺癌细胞MCF-7的时间毒性

三氯生(triclosan,TCS)是一种广谱性抗菌剂,2005年欧盟水框架指令将TCS列为一种新型污染物。目前对TCS的研究局限于急性毒性实验,关于TCS毒性随时间的变化以及不同溶解状态下TCS的毒性差异的研究却鲜有报道。应用以96孔微板为暴露反应载体的微板毒性分析法,添加氢氧化钠(NaOH)或使用二甲亚砜(DMSO)作为助溶剂溶解TCS,分别测定其对青海弧菌Q67的相对发光抑制毒性(15min急性毒性和时间毒性)和对人乳腺癌细胞MCF-7在不同暴露时间(24、48和72h)内的细胞增殖抑制毒性。Q67的急性毒性实验结果表明,碱性条件下TCS的毒性(EC50=3.97(10-8mol.L-1)大于DMSO作为助溶剂时的毒性(EC50=1.68(10-4mol.L-1)。无论碱性条件还是DMSO助溶,TCS在不同暴露时间内对Q67的时间毒性没有明显差异。在不同暴露时间下MCF-7增殖抑制率实验中,DMSO作为助溶剂时,TCS的最高实验浓度为1.46(10-3mol.L-1,随着暴露时间的延长,抑制率在24、48和72h时分别为27.8%、44.2%和62.4%;碱性环境时TCS的最高实验浓度为1.39(10-6mol.L-1,随着暴露时间的延长,抑制率在24、48和72h时分别为20.2%、55.8%和73.9%。研究表明,在DMSO和NaOH作为助溶剂的条件下,TCS对MCF-7均存在时间毒性差异,并且NaOH碱性溶液中TCS对MCF-7的毒性远大于DMSO作为助溶剂时的毒性。     

A Computational Fluid Dynamics Approach for Urban Area Transport and Dispersion Modeling

A simulation tool has been developed to model the wind fields, turbulence fields, and the dispersion of Chemical, Biological, Radiological and Nuclear (CBRN) substances in urban areas on the building to city blocks scale. A Computational Fluid Dynamics (CFD) approach has been taken that naturally accounts for critical flow and dispersion processes in urban areas, such as channeling, lofting, vertical mixing and turbulence, by solving the steady-state, Reynolds-Averaged Navier–Stokes (RANS) equations. Rapid generation of high quality cityscape volume meshes is attained by a unique voxel-based model generator that directly interfaces with common Geographic Information Systems (GIS) file formats. The flow and turbulence fields are obtained by solving the steady-state RANS equations using a collocated, pressure-based approach formulated for unstructured and polyhedral mesh elements. Turbulence modeling is based upon the Renormalization Group variant of the k–ε model (k–ε RNG). Neutrally buoyant simulations are made by prescribing velocity boundary condition profiles found by a power–law relationship, while turbulence quantities boundary conditions are defined by a prescribed mixing length in conjunction with the assumption of turbulence equilibrium. Dispersion fields are computed by solving an unsteady transport equation of a dilute gas, formulated in a Eulerian framework, using the velocity and turbulence fields found from the steady-state RANS solution. In this paper the model is explained and detailed comparisons of predicted to experimentally obtained velocity, turbulence and dispersion fields are made to neutrally stable wind tunnel and hydraulic flume experiments.     

Beyond eddy diffusivity: an alternative model for turbulent dispersion

Turbulent dispersion proceeds not only much faster but also in a qualitatively different manner than molecular diffusion. Yet, the majority of hydraulic, oceanic and atmospheric models rely on the concept of an eddy diffusivity. It is shown here that an alternative model can be developed to exhibit observed behavior. The new term in the diffusion equation, which is non-local, may be interpreted in terms of the probability density function (pdf) of the turbulent velocity. Different assumptions about this distribution lead to a family of models, one of which is the model proposed here and another, the classical Fickian model of diffusion. A connection is also made with models using fractional calculus.     

Anti-diffusion and source identification with the ‘RAW’ scheme: A particle-based censored random walk

This paper presents a novel methodology for time reversal in advective-diffusive pollutant transport in groundwater systems and other environmental flow systems (specifically: time reversal of diffusive terms). The method developed in this paper extends previous particle-based approaches like the Reversed Time Particle Tracking Method of Bagtzoglou [6]. The reversal of the ‘diffusive’ and/or ‘macrodispersive’ component of pollutant migration is especially under focus here. The basis of the proposed scheme for anti-diffusion is a continuous time, censored, non-local random walk capable of tracking groundwater solute concentration profiles over time while conserving the (reverse) Fickian properties of the anti-diffusing particle cloud in terms of moments. This scheme is an alternative to the direct solution of the eulerian concentration-based diffusion PDE, which is notoriously unstable in reverse time. Our analysis leads to the conclusion that an adaptive time stepping scheme—with decreasing time step—is necessary in order to maintain a constant amount of anti-diffusion (the reverse form of Fick’s law). Specifically, we study the relations between the following parameters: time step evolution vs. time (or vs. number of steps); variance evolution (decrease rate); total time (or number of steps) required to reach a fully anti-diffused solution. The proposed approach is shown to be quite efficient; typically, for every ten time steps, one to two orders of magnitude reduction of the dispersion width of the plume can be attained. Furthermore, the method is shown to be asymptotically exact for reverse Fickian diffusion. The method is applied with success to several situations involving the diffusive transport of a conservative solute in the following cases: (i) Single source recovery in one-dimensional space with constant diffusion parameters (this example serves as a validation test for the theory); (ii) Single source recovery in two-dimensional space with constant isotropic diffusion (this example also serves as a test for the theory); (iii) Multiple source recovery in two-dimensional space, assuming isotropic diffusion. It is expected that the methodology tested in this paper is applicable more generally to complex environmental pollution problems involving multiple sources, anisotropic hydrodynamic dispersion, and space-time variable advection-diffusion flow systems; the modeling of reverse diffusion/dispersion in such systems is currently under investigation.     

Development of an empirical equation for the transverse dispersion coefficient in natural streams

In this study, a new empirical equation for the transverse dispersion coefficient has been developed based on the hydraulic and geometric parameters in natural streams using a regression technique. First, a total of 32 data sets in 16 streams were collected. Among those sets, 16 sets were used for deriving the new equation, and the other 16 sets were used for verifying the equation. Then, through dimensional analysis, it was found that the normalized transverse dispersion coefficient is associated with several parameters such as sinuosity, aspect ratio, and a friction term. The robust least square method was applied to estimate regression coefficients. The newly proposed equation was proven to be superior in explaining the dispersion characteristics of natural streams more precisely compared to the existing equations.     

A Numerical Study for Turbulent Flow and Thermal Influence over Inhomogenous Canopy of Roughness Elements

A large-eddy simulation with transitional structure function(TSF) subgrid model we previously proposed was performed to investigate the turbulent flow with thermal influence over an inhomogeneous canopy, which was represented as alternative large and small roughness elements. The aerodynamic and thermodynamic effects of the presence of a layer of large roughness elements were modelled by adding a drag term to the three-dimensional Navier–Stokes equations and a heat source/sink term to the scalar equation, respectively. The layer of small roughness elements was simply treated using the method as described in paper (Moeng 1984, J. Atmos Sci. 41, 2052–2062) for homogeneous rough surface. The horizontally averaged statistics such as mean vertical profiles of wind velocity, air temperature, et al., are in reasonable agreement with Gao et al.(1989, Boundary layer meteorol. 47, 349–377) field observation (homogeneous canopy). Not surprisingly, the calculated instantaneous velocity and temperature fields show that the roughness elements considerably changed the turbulent structure within the canopy. The adjustment of the mean vertical profiles of velocity and temperature was studied, which was found qualitatively comparable with Belcher et al. (2003, J Fluid Mech. 488, 369–398)’s theoretical results. The urban heat island(UHI) was investigated imposing heat source in the region of large roughness elements. An elevated inversion layer, a phenomenon often observed in the urban area (Sang et al., J Wind Eng. Ind. Aesodyn. 87, 243–258)’s was successfully simulated above the canopy. The cool island(CI) was also investigated imposing heat sink to simply model the evaporation of plant canopy. An inversion layer was found very stable and robust within the canopy.     

Turbulent secondary flows in wall turbulence: vortex forcing,scaling arguments,and similarity solution

Spanwise surface heterogeneity beneath high-Reynolds number, fully-rough wall turbulence is known to induce a mean secondary flow in the form of counter-rotating streamwise vortices—this arrangement is prevalent, for example, in open-channel flows relevant to hydraulic engineering. These counter-rotating vortices flank regions of predominant excess(deficit) in mean streamwise velocity and downwelling(upwelling) in mean vertical velocity. The secondary flows have been definitively attributed to the lower surface conditions, and are now known to be a manifestation of Prandtl’s secondary flow of the second kind—driven and sustained by spatial heterogeneity of components of the turbulent (Reynolds averaged) stress tensor (Anderson et al. J Fluid Mech 768:316–347, 2015). The spacing between adjacent surface heterogeneities serves as a control on the spatial extent of the counter-rotating cells, while their intensity is controlled by the spanwise gradient in imposed drag (where larger gradients associated with more dramatic transitions in roughness induce stronger cells). In this work, we have performed an order of magnitude analysis of the mean (Reynolds averaged) transport equation for streamwise vorticity, which has revealed the scaling dependence of streamwise circulation intensity upon characteristics of the problem. The scaling arguments are supported by a recent numerical parametric study on the effect of spacing. Then, we demonstrate that mean streamwise velocity can be predicted a priori via a similarity solution to the mean streamwise vorticity transport equation. A vortex forcing term has been used to represent the effects of spanwise topographic heterogeneity within the flow. Efficacy of the vortex forcing term was established with a series of large-eddy simulation cases wherein vortex forcing model parameters were altered to capture different values of spanwise spacing, all of which demonstrate that the model can impose the effects of spanwise topographic heterogeneity (absent the need to actually model roughness elements); these results also justify use of the vortex forcing model in the similarity solution.     

A mathematical model for type II profile of concentration distribution in turbulent flows

This paper presents a mathematical model to investigate type II profile of suspension concentration distribution (i.e., the concentration profile where the maximum concentration appears at some distance above the bed surface) in a steady, uniform turbulent flow through open-channels. Starting from the mass and momentum conservation equations of two-phase flow, a theoretical model has been derived. The distribution equation is derived considering the effects of fluid lift force, drag force, particle inertia, particle–particle interactions, particle velocity fluctuations and drift diffusion. The equation is solved numerically and is compared with available experimental data as well as with other models existing in the literature. Good agreement between the observed value and computed result, and minimum error in comparison to other models indicate that the present model can be applied in predicting particle concentration distribution for type II profile for a wide range of flow conditions. The proposed model is also able to show the transition from type I profile to type II profile.     

Structure of turbulent plumes from a momentumless source in a smooth bed

The spatial development of a passive scalar plume is studied within the inhomogeneous turbulence of a boundary layer flow in a recirculating laboratory flume with a smooth bed. The source of the scalar is located flush with the bed, and the low-momentum source design is intended to simulate a diffusive-type scalar release. A weakly diffusive fluorescent dye is used as the scalar. Planar laser-induced fluorescence (PLIF) techniques were used to record the structure of the plume at a spatial resolution of 150 μm. The measured structure of the mean concentration field is compared to an analytical solution for shear-free, homogeneous turbulence. The laboratory plume exhibits spatial development in the mean concentration field that deviates from the self-similar behavior predicted by the analytical solution; this deviation is due to the mean shear and inhomogeneity of the turbulence. In particular, the influence of the viscous sublayer on the plume development is seen to be significant. Nonetheless, the analytical solution replicates some of the features seen in the laboratory plume, and the solution suggests methods of reducing the laboratory data even for cases where the results deviate from the analysis. We also examine the spatial development of the root-mean-square (rms) fluctuating concentration field, and use scalar probability density functions to examine the relationship between the mean and fluctuating concentrations.     

A new model for soil–water drainage problems

Seepage flow is an agent related to the transport and dispersion of contamination in groundwater. Steady two-dimensional seepage flow is governed by Laplace’s equation, for which several solution techniques are available. Because computations are complex from a practical point of view, simplified models encompass the Dupuit-Forchheimer approach assuming a horizontal flow. However this approach is inaccurate in seepage problems involving steep drawdowns. In this research, a new theoretical model for 2D seepage flow is proposed based on Fawer’s theory for curved flows Castro-Orgaz (Environ Fluid Mech 10(3):2971–2310, 2010), from which a second-order equation results describing the seepage surface. From this development, a numerical solution for the rectangular dam problem based on the second-order model is presented, whereas a simple first-order equation is found to describe flow to drains under a uniform rainfall. The results of this new model are compared with the full 2D solution of Laplace’s equation for typical test cases, resulting in an excellent agreement.     

Improving the effectiveness of angular dispersion in plant neighbourhood models

Spatial arrangement can be an important factor affecting competition among plants. We evaluated three ways to improve the effectiveness of angular dispersion (AD) for describing spatial arrangement in plant neighbourhood models. First, we modified Zar's (1974) AD formula by weighting each neighbour by its competitive influence. We calculated this using two different competition indices to derive an AD of competitive influence, rather than of equally weighted plant locations, around a subject plant. Secondly, we constrained the effect of AD on the neighbourhood model using an optimised parameter that defines the minimum value that AD can adopt. Thirdly, we included the direction in which competition is concentrated (the mean azimuth of the weighted AD) in the growth models. These developments were evaluated within a radial growth model of Scots pine and birch growing in semi-natural, spatially heterogeneous forest. Weighted AD resulted in significant improvements in predicted radial growth of target trees over the traditional measure of AD. The optimised parameter that defines the minimum value of AD consistently evolved values significantly higher than zero. This suggests that clumped and dispersed neighbourhoods do not differ in their negative effects on a subject tree as much as expected. The inclusion of directional components of the weighted AD did not improve the accuracy of the growth models. Weighting of the angular dispersion of neighbours improved the performance of local competition models.     

Appraisal of salinity and fluoride in a semi-arid region of India using statistical and multivariate techniques

Various physico-chemical parameters, including fluoride (F⁻), were analyzed to understand the hydro-geochemistry of an aquifer in a semi-arid region of India. Furthermore, the quality of the shallow and deep aquifer (using tube well and hand pumps) was also investigated for their best ecological use including drinking, domestic, agricultural and other activities. Different multivariate techniques were applied to understand the groundwater chemistry of the aquifer. Findings of the correlation matrix were strengthened by the factor analysis, and this shows that salinity is mainly caused by magnesium salts as compared to calcium salts in the aquifer. The problem of salinization seems mainly compounded by the contamination of the shallow aquifers by the recharging water. High factor loading of total alkalinity and bicarbonates indicates that total alkalinity was mainly due to carbonates and bicarbonates of sodium. The concentration of F⁻ was found more in the deep aquifer than the shallow aquifer. Further, only a few groundwater samples lie below the permissible limit of F⁻, and this indicates a risk of dental caries in the populace of the study area. The present study indicates that regular monitoring of groundwater is an important step to avoid human health risks and to assess its quality for various ecological purposes.