Instantaneous transport of a passive scalar in a turbulent separated flow

The results of large-eddy simulations of flow and transient solute transport over a backward facing step and through a 180° bend are presented. The simulations are validated successfully in terms of hydrodynamics and tracer transport with experimental velocity data and measured residence time distribution curves confirming the accuracy of the method. The hydrodynamics are characterised by flow separation and subsequent recirculation in vertical and horizontal directions and the solute dispersion process is a direct response to the significant unsteadiness and turbulence in the flow. The turbulence in the system is analysed and quantified in terms of power density spectra and covariance of velocity fluctuations. The injection of an instantaneous passive tracer and its dispersion through the system is simulated. Large-eddy simulations enable the resolution of the instantaneous flow field and it is demonstrated that the instabilities of intermittent large-scale structures play a distinguished role in the solute transport. The advection and diffusion of the scalar is governed by the severe unsteadiness of the flow and this is visualised and quantified. The analysis of the scalar mass transport budget quantifies the mechanisms controlling the turbulent mixing and reveals that the mass flux is dominated by advection.     

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.     

Solution of fully-coupled shallow water equations and contaminant transport using a primitive-variable Riemann method

A Riemann-solver scheme, using primitive variables rather than conserved variables, is configured and tuned for the solution of the fully-coupled two-dimensional shallow water and contaminant transport equations. This scheme is based on the unstructured finite volume discretization using primitive-variable Roe-flux approximation with an entropy fix. The primitive-variable flux associated with the exact source-term balancing is well-behaved and well-balanced for both still-water and dry regions with arbitrary bed topography. Second-order accuracy is used in space and time. The present study uses a nonlinear implicit scheme based on Newton-iterative algorithm for the time integration. In order to show the accuracy of the scheme, numerical results are verified by different test cases for contaminant advection and diffusion. A scenario of contaminant transport in a complex geometry with wet and dry elements is also simulated to demonstrate that the present work can be implemented on practical applications involving flooding and contaminant transport.     

The wave-induced solute flux from submerged sediment

The issue of the transport of dissolved nutrients and contaminants between the sediment in the bottom of a lake or reservoir and the body of water above it is an important one for many reasons. In particular the biological and chemical condition of the body of water is intricately linked to these mass transport processes. As the review by Boudreau (Rev Geophys 38(3):389–416, 2000) clearly demonstrates those transport processes are very complex involving mechanisms as diverse as the wave-induced flux between the sediment and the overlying water and the effect of burrowing animals on the transport within the sediment as well as basic diffusion mechanisms. The present paper focuses on one facet of these transport processes; we re-examine the balance of diffusion and wave-induced advection and demonstrate that the wave-induced flux of a solute from submerged sediment is not necessarily purely diffusive as suggested by Harrison et al. (J Geophys Res 88:7617–7622, 1983) but can be dominated by a mean or time-averaged flux induced by the advective fluid motion into and out of the sediment caused by the fluctuating pressure waves associated with wave motion. Indeed along the subtidal shoreline where the fluctuating bottom pressures are greatest, wave-induced advection will dominate the mean, time-averaged transport of solute into or out of the sediment as suggested in the work of Riedl et al. (Mar Biol 13:210–221, 1972). However, the present calculations also indicate that this advective flux decreases rapidly with increasing depth so that further away from the shoreline the advective flux becomes negligible relative to the diffusive flux and therefore the latter dominates in deeper water.     

Solute Transport in Open-Channel Networks in Unsteady Flow Regime

In this paper we propose a robust algorithm to evaluate solute transport in open-channel networks with transient storage under an unsteady flow regime. In the proposed approach, through the integration of junction equations into the model and solving them explicitly, the analysis of solute transport problems in open-channel networks is simplified significantly. Furthermore, when coupled with a transient hydrodynamic open-channel network model for flow simulation, the proposed model can be utilized in the solution of solute transport problems under unsteady flow regimes. In the proposed model, the governing equations are written in a conservative form and are solved using a fractional-step algorithm, which is based on a relaxation and central difference scheme. The proposed algorithm is robust and accurate even for advection dominant cases. A pure advection with discontinuities, a field application and solute transport in an open-channel network in an unsteady flow regime are included, to demonstrate the performance of the proposed algorithm.     

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.     

A σ-coordinate transport model coupled with rotational boussinesq-type equations

This paper describes a σ-coordinate scalar transport model coupled with a Boussinesq-type hydrodynamic model. The Boussinesq model has the ability to calculate both three-dimensional velocity distributions and the water surface motion. To capture ‘dispersion’ processes in open channel flow, horizontal vorticity effects induced by a bottom shear stress are included in the Boussinesq model. Thus, a reasonable representation of vertical flow structure can be captured in shallow and wavy flow fields. To solve the coupled Boussinesq and scalar transport system, a finite-volume method, based on a Godunov-type scheme with the HLL Riemann solver, is employed. Basic advection and advection–diffusion numerical tests in a non-rectangular domain were carried out and the computed results show good agreement with analytic solutions. With quantitative comparisons of dispersion experiments in an open channel, it is verified that the proposed coupled model is appropriate for both near and far field scalar transport predictions. From numerical simulations in the surf zone, physically reasonable results showing expected vertical variation are obtained.     

Eddy covariance CO2 flux measurements in nocturnal conditions: an analysis of the problem

A detailed analysis of the various processes at work in stable boundary layers was made. It pointed out that two main mechanisms may affect eddy covariance measurements in stable conditions and that their impacts were different. On one hand, intermittent turbulence produces strongly nonstationary events during which the validity of turbulent transport and storage measurements is uncertain. On the other hand, during breeze and drainage flow events, significant advection takes place and competes with turbulent flux and storage. Intermittent turbulence questions both the ability of eddy covariance systems to adequately capture turbulent flux and storage and the representativeness of the measurements. Ability of the systems to capture the fluxes could be improved by adapting the averaging time period or the high pass filtering characteristics. However, beyond this, the question of representativeness of the flux remains open as the flux measured during an intermittent turbulence event represents not only the source term, but also the removal of CO2 that built up in the control volume and that cannot be simply related to the source term. In these conditions, the u* discrimination is likely to be insufficient and should be completed with a stationarity criterion. Further research should allow determining better selection criteria. Advection occurs mainly in presence of flows associated with topographical slopes (drainage flows) or with land use changes (breezes). Direct advection measurements were performed at several sites, but the results were shown to be strongly site dependent. A classification based on the general flow pattern and on the source intensity evolution along streamlines was proposed here. Five different patterns were identified that helped to classify the different observations. The classification was found to be a fairly good fit for the observations. This could serve as a tool to better understand and quantify the fluxes at sites subjected to repeatable patterns.     

Calibration and predictive ability analysis of longitudinal solute transport models in mountain streams

The first step in developing travel time and water quality models in streams is to correctly model solute transport mechanisms. In this paper a comparison between two solute transport models is performed. The parameters of the Transient Storage model (TS) and the Aggregated Dead Zone model (ADZ) are estimated using data of thirty seven tracer experiments carried out under different discharges in five mountain streams of Colombian Los Andes. Calibration is performed with the generalized uncertainty estimation method (GLUE) based on Monte-Carlo simulations. Aspects of model parameters identifiability and model parsimony are analyzed and discussed. The TS model with four parameters shows excellent results during calibration but the model parameters present high interaction and poor identifiability. The ADZ model with two independent and clearly identifiable parameters gives sufficiently precise calibration results. As a conclusion, it is stated that the ADZ model with only two parameters is a parsimonious model that is able to represent solute transport mechanisms of advection and longitudinal dispersion in the studied mountain streams. A simple model parameter estimation methodology as a function of discharge is proposed in this work to be used in prediction mode of travel time and solute transport applications along mountain streams.     

Solute transport in a semi-infinite homogeneous aquifer with a fixed point source concentration

This study derives an analytical solution of the advection–dispersion (AD) equation commonly used to describe the transport of pollutants in a semi-infinite homogeneous aquifer. When an extra constant source term is added to the AD equation, it changes the solution of the equation. The AD equation is solved analytically using Laplace transform. Also, the equation is solved numerically using an explicit finite difference method and its stability condition is presented with the aid of matrix method. For the solution of the AD equation the following considerations are made: (1) The dispersion and velocity are considered as time-dependent; (2) dispersion is expressed as directly proportional to the square of velocity; (3) there is also diffusion; (4) there is some initial concentration and the aquifer domain is, therefore, not pollutant-free; (5) There is a time-dependent exponentially decreasing input source; and (6) the concentration gradient is assumed to be zero at the exit boundary. It is found that the contaminant concentration decreases with time contrary to what happens when the extra term is not included.     

Large-eddy simulation of low-level jet-like flow in a canopy

The computational method of Large-Eddy Simulations has been used to study the weak, neutrally stable drainage flow within tree canopies. The computational results show that a secondary velocity maximum that resembles a jet is formed within the canopy under the nocturnal flow conditions. This jet-like flow is important in the analysis and measurements of the net ecosystem-atmosphere exchange (NEE) for carbon dioxide (CO₂). A uniformly distributed, plane source was placed within the canopy in order to simulate the nocturnal production of CO₂. The NEE is calculated as the sum of the integration of the rate of change of the concentration of CO₂ over the computational domain, the vertical turbulent flux measured directly by eddy-covariance (EC) method, and the advection terms, which are not taken into account in the EC method. Numerical results of the velocity and concentration fields, within and above the canopy, are presented and their impact on the CO₂ transport is investigated in detail. The computational results show that 15–20% of NEE is drained out by the advection process under the canopy. The results also show that the turbulent fluctuations in the lateral direction are also significant and may result in 2–5% CO₂ transport.     

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.     

Adsorption distribution impact on preferential transport within horizontal flow constructed wetland (HFCW)

Predicting preferential water flow and solute transport in soils is in the interest of scientists and engineers in the fields of agricultural soil, forest hydrology, soil physics and wastewater treatment by constructed wetland. In artificial wetlands, any preferential pathway induces an insufficient residence time of pollutants in the soil, making an incomplete and unfinished biodegradation process, a wrong evaluation of the hydraulic residence time of the system which would hinder its management in a complete system with several entities of treatment and a non-homogeneous growth of the biofilm in the solid filter mass. This paper is a contribution in tracer experiment data analysis within a horizontal flow constructed wetland built in a storm water basin. A two-dimensional numerical model was used to simulate flow and reactive solute transport. The flow model was successfully calibrated in very dry soil conditions. The adsorption profiles used in the reactive transport modeling are those of five molecules: metolachlor, atrazine, deethylatrazine (DEA), deisopropylatrazine (DIA), and hydroxyatrazine (HA). We show that the adsorption distribution is an internal factor of soil which is responsible to the “preferential” pathway transport in a homogeneous gravel texture. The mean residence time of pollutants within the filter is strongly correlated with the average value of the adsorption coefficient. Moreover, we note a lack of significant impact of the heterogeneity of the medium on the statistical moments of breakthrough curve. It appears that a uni-modal breakthrough curve is not significant to the absence of preferential flow in the medium and at least a two-dimensional display can provide sufficient evidence on the presence or absence of preferential pathways. Finally, using a PLS regression, it is possible to perfectly discriminate the number of peaks of concentration and the asymmetry of the breakthrough curve.     

Characteristic volume fractions of different grains in porous media for anomalous dispersion

Anomalous solute transport in porous media is an important issue in groundwater research. In this paper, we explore the relationship between the anomalous solute transport and the volume fractions of different grains in the porous media. Via simulation, we find that there is a maximum and a minimum in the degree of anomalous transport when changing the volume fractions of different grains. Moreover, the characteristic volume fractions corresponding to the anomalous transport maximum and minimum vary little with the flow field and diffusion coefficient of the solute. We also find that the volume fraction corresponding to the most anomalous dispersion is close to the threshold of the site percolation for simple-cubic networks.     

Advection and resulting CO2 exchange uncertainty in a tall forest in central Germany

Potential losses by advection were estimated at Hainich Forest, Thuringia, Germany, where the tower is located at a gentle slope. Three approaches were used: (1) comparing nighttime eddy covariance fluxes to an independent value of total ecosystem respiration by bottom-up modeling of the underlying processes, (2) direct measurements of a horizontal CO2 gradient and horizontal wind speed at 2 m height in order to calculate horizontal advection, and (3) direct measurements of a vertical CO2 gradient and a three-dimensional wind profile in order to calculate vertical advection. In the first approach, nighttime eddy covariance measurements were compared to independent values of total ecosystem respiration by means of bottom-up modeling of the underlying biological processes. Turbulent fluxes and storage term were normalized to the fluxes calculated by the bottom-up model. Below a u(*) threshold of 0.6 m/s the normalized turbulent fluxes decreased with decreasing u(*), but the flux to the storage increased only up to values less than 20% of the modeled flux at low turbulence. Horizontal advection was measured by a horizontal CO2 gradient over a distance of 130 m combined with horizontal wind speed measurements. Horizontal advection occurred at most of the evenings independently of friction velocity above the canopy. Nevertheless, horizontal advection was higher when u(*) was low. The peaks of horizontal advection correlated with changes in temperature. A full mass balance including turbulent fluxes, storage, and horizontal and vertical advection resulted in an increase of spikes and scatter but seemed to generally improve the results from the flux measurements. The comparison of flux data with independent bottom-up modeling results as well as the direct measurements resulted in strong indications that katabatic flows along the hill slope during evening and night reduces the measured apparent ecosystem respiration rate. In addition, anabatic flows may occur during the morning. We conclude that direct measurements of horizontal and vertical advection are highly necessary at sites located even on gentle hill slopes.     

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.     

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.     

CO2 fluxes near a forest edge: a numerical study

In contrast with recent advances on the dynamics of the flow at a forest edge, few studies have considered its role on scalar transport and, in particular, on CO2 transfer. The present study addresses the influence of the abrupt roughness change on forest atmosphere CO2 exchange and contrasts the concentration and flux fields against those of a uniform forested surface. We use an atmospheric boundary layer two-equation closure model that accounts for the flow dynamics and vertical divergence of CO2 sources/sinks within a plant canopy. This paper characterizes the spatial variation of CO2 fluxes as a function of both sources/sinks distribution and the vertical structure of the canopy. Results suggest that the ground source plays a major role in the formation of wave-like vertical CO2 flux behavior downwind of a forest edge, despite the fact that the contribution of foliage sources/sinks changes monotonously. Such a variation is caused by scalar advection in the trunk space and reveals itself as a decrease or increase in vertical fluxes over the forest relative to carbon dioxide exchange of the underlying forest. The effect was more pronounced in model forests where the leaf area is concentrated in the upper part of the canopy. These results can be useful both for interpretation of existing measurements of net ecosystem exchange of CO2 (NEE) from flux towers in limited fetch conditions and in planning future CO2 transport experiments.     

Deterministic particle tracking simulation of pollutant discharges in rivers and estuaries

A two-dimensional deterministic particle tracking model, in which the anisotropic-dispersive process is described by a particle strength exchange scheme, is established for the simulation of pollutant transport in vertically well-mixed rivers and estuaries. By simulating two benchmark problems with analytic solutions, the PSE scheme is shown to be accurate even if the anisotropic ratio of dispersion coefficients is very high. Further simulations of two specific problems concerning the optimal effluent discharge location and procedure are presented. The major conclusion is that in a tidal estuary with a relatively large fresh-water flow, setting the discharge position at the mixing center and making the discharge rate proportional to flow speed may minimize the peaks of concentration.