Transport in a three-zone wetland: Flow velocity profile and environmental dispersion

doi:10.1016/j.cnsns.2014.04.025

Communications in Nonlinear Science and Numerical Simulation

Volume 20, Issue 1, January 2015, Pages 136-153

https://doi.org/10.1016/j.cnsns.2014.04.025 Get rights and content

Highlights

•
To find the velocity profile of the superficial flow through a three-zone wetland.
•
To determine the environmental dispersivity by a multi-scale analysis.
•
To analyze the effects of dimensionless characteristic parameters on the velocity profile and environmental dispersivity.
•
To illustrate the difference of dispersivity between the three-zone flow and three-layer flow.

Abstract

To achieve better understanding of transport process in real waterways interacted with adjacent aquatic vegetation and riparian buffers, a three-zone model featuring the bank-effect is presented to character the water flow and environmental dispersion. Based on basic formulation in context of porous media flow, the velocity profile of a fully developed flow through the wetland is derived, with that for single zone and two-zone wetland flows recovered as special cases. The environmental dispersivity is determined by the approach of multi-scale analysis, with the effects of dimensionless parameters well illustrated. Application examples are provided to illustrate associated hierarchical structure for the critical length and duration of the contaminant cloud, and a comparison is made between the three-zone wetland and a corresponding three-layer wetland.

Introduction

Wetlands can provide significant ecological services including water supply, water quality improvement, wildlife conservation, climate regulation, etc. [1], [2], [3]. For ecological risk assessment, ecological restoration and wastewater treatment engineering associated with wetlands, either natural or constructed, a fundamental issue is the contaminant transport characterized by environmental dispersion [4], [5], [6], [7], [8].

Numerous endeavors have been contributed to understanding the mechanisms of solute dispersion in water bodies since the reputable work of [9]. The concept of Taylor dispersion refers to the process that dissolved contaminant disperses towards longitudinal direction under the combined action of lateral solute diffusion and the transverse non-uniformity of the longitudinal velocity [9], and it has been extended for a variety of environmental flows including those in wetlands, rivers and estuaries [4], [10], [11], [12], [13], [14], [15]. Elder investigated the dispersion in a channel of constant depth, which revealed that both the depth and shear velocity contribute to the dispersion coefficient [10]. Fischer analyzed the mechanisms of transporting the dissolved constituents in oscillatory, estuary flow, and found that the dispersion coefficient can sometimes be negative [16]. Guymer conducted a series of laboratory experiments on a large scale channel with sinuous planform geometry, indicating that the more ‘natural’ cross-sectional geometry channel exhibits strongly skewed distributions associated with an increase in the value of a longitudinal dispersion parameter [17]. Kashefipour and Falconer developed an equation, which relates dispersion coefficient to the hydraulic and geometric parameters of the flow, for predicting the longitudinal dispersion coefficient in riverine flows [18]. A technique for predicting the longitudinal dispersion coefficient is proposed by Seo and Baek based on the beta function for the transverse velocity distribution in natural streams [19]. Wu and Chen recently explored the mechanism of approach toward transverse uniformity of concentration distribution, providing new physical insight into Taylor dispersion [20], [21]. However, these researches were conducted for pure fluid flows, without the effect of aquatic vegetation on dispersion into consideration.

Ubiquitous in rivers, estuaries and wetlands, vegetation strongly influence water conveyance and mass transport, introducing more complex mechanisms to the dispersion process [22]. J $\ddot{a}$ rvel $\ddot{a}$ carried out flume studies to investigate flow structure around flexible vegetation, the result of which showed that the flow above the wheat reasonably followed the log law [23]. Lightbody and Nepf’s work clarified that profiles of vertical velocities within emergent vegetation are directly influenced by the distribution of the vegetation elements [24]. For the emergent and submerged natural vegetation in open channel flow, Shucksmith explored the longitudinal dispersion coefficients [25]. The above experimental and field measurement researches were mainly limited to the mean and turbulent flow and mass transport in the presence of aquatic vegetation, associated with micro-dispersion in terms of phase average based superficial diffusion as Taylor dispersion in the micro-flow in the porous media. However, in the environmental area, what we concern most is the Taylor dispersion at the environmental scale, namely the environmental dispersion, which is a kind of macro-dispersion out of standard environmental and risk assessment. Attributing to the presence of aquatic vegetation, the micro-dispersion becomes more complex, inevitably resulting in different apparent characteristics of macro-dispersion at the environmental scale. Both the micro-dispersion and macro-dispersion are affiliated to Taylor dispersion, but the key points are the distinctiveness of the involved scales. In context of mechanics for porous media flows, Chen and his collaborators defined and explored the environmental dispersion in terms of porous media in single-channel wetland flows dominated by the depth-, width-, and depth-and-width-effects, using Aris’ method of concentration moment or the method of multi-scale analysis [15], [26], [27], [28], [29], [30], [31]. By considering the effect of inhomogeneous distribution of vegetation in wetlands, Wu et al. [32] explored the environmental dispersion in wetlands with two zones that are distinctly vegetated, resorting to the approach of concentration moments. As the extension of the modeling on contaminant spreads in a single-layer wetland with homogeneous vegetation [27], Chen et al. [4] analyzed the environmental dispersion in two-layer wetlands with layered vegetation. The critical length and duration of the contaminant cloud are presented, which give apparent results of dispersion at the concerned environmental scale.

Vegetation occupied in wetlands is mostly characterized by different canopy and stem morphology. Nepf [7] made filed and experimental observations for the flow and mass transport in the presence of vegetation at the micro-scale. The characteristics of velocity and turbulent in micro-region inevitably bring influences on the apparent environmental dispersivity at the concerned environmental scale. As an analytical effort, Wang et al. [33] investigated the flow and environmental dispersion in a three-layer wetland dominated by free-water-surface-effect. By Aris’s method of concentration moments, the asymptotic variation of the dispersivity is concretely illustrated.

Waterways are highly interactive with the adjacent riparian buffers [34]. Typical examples are wet meadows dominated by herbaceous plants that are usually water-emergent or water-submergent on either side of the flow [35], and the river along with riparian vegetation during overbank flows in the rainy season [36]. It is well documented that vegetated riparian zones can strongly influence the hydraulics of the entire channel due to the invasion of gravels and stems [36]. Riparian vegetation consequently strengthens the flow in the resulting narrowed channel. An awareness of the flow and dispersion under the effect of multi-channel and spatial-structural vegetation in the wetland is essential for understanding the transport of pollutants in open channel flows. Different from the three-layer model featuring the wetlands dominated by layered vegetation and characterized by free-surface effect, a three-zone model is devised to represent the typical case with three zones occupied with heterogeneous vegetation (though homogeneous and in each zone), as shown in Fig. 1.

With the presentation of a three-zone wetland model, the specific objectives of this paper are: (a) to derive and illustrate the flow velocity profile; (b) to present a longitudinal transport model for environmental dispersion; (c) to determine the environmental dispersivity by an intuitive technique of multi-scale analysis instead of the complicated calculus of concentration moments; (d) to analyze the effects of dimensionless characteristic parameters on the velocity profile and environmental dispersivity, and (e) to find applications of the model in the description of hierarchical structure for the critical length and duration of the contaminant cloud.

Section snippets

Formulation for momentum transport and velocity profile

For a typical wetland flow, the basic equation for momentum transport can be adopted generally at the phase average scale as [27], [37], [38] $ρ (\frac{\partial U}{\partial t} + \nabla \cdot \frac{UU}{ϕ}) = - \nabla P - μ F U + κ μ \nabla^{2} U + κ \nabla \cdot (L \cdot \nabla U),$ where $ρ$ is the density $[kg m^{- 3}], U$ velocity $[m s^{- 1}], t$ time $[s], ϕ$ porosity $[dimensionless], P$ pressure $[kg m^{- 1} s^{- 2}], μ$ dynamic viscosity $[kg m^{- 1} s^{- 1}], F$ shear factor $[m^{- 2}], κ$ tortuosity $[dimensionless]$ to account for the spatial structure of aquatic plants, $L$ momentum dispersivity tensor $[kg m^{- 1} s^{- 1}]$ .

Consider momentum transport in a

Formulation for concentration transport

The basic equation for the transport of the superficial concentration in a wetland flow can be adopted as [27], [37], [38] $ϕ \frac{\partial C}{\partial t} + \nabla \cdot (U C) = \nabla \cdot (κ λ ϕ \nabla C) + κ \nabla \cdot (K \cdot \nabla C),$ where C concentration $[kg m^{- 3}], λ$ concentration diffusivity $[m^{2} s^{- 1}]$ , and $K$ concentration dispersivity tensor $[m^{2} s^{- 1}]$ .

Consider a uniform and instantaneous release of contaminant with mass Q at the cross-section of $x = 0$ at time $t = 0$ , the initial conditions can be set as ${C_{k} (x, y_{k}, t)|}_{t = 0} = \frac{Q δ (x)}{ϕ_{k} (W)}, k = 1, 2, 3,$ where $δ (x)$ is the Dirac delta function, $C_{1}, C_{2}$ and

Multi-scale analysis for concentration dispersion

With dimensionless parameters of $ξ = \frac{x}{L}, ψ_{k} = \frac{u_{k}}{〈u〉}, τ = \frac{t}{L / 〈u〉},$ where the transverse average operator for a variable f is defined as $〈f〉 = \int_{0}^{r} f_{1} d ζ_{1} + \int_{r}^{θ} f_{2} d ζ_{2} + \int_{θ}^{1} f_{3} d ζ_{3},$ the governing equations for concentration corresponding boundary and initial conditions can be rewritten as $ε \cdot \frac{\partial C_{k}}{\partial τ} + ψ_{ϕ k} \cdot ε \cdot \frac{\partial C_{k}}{\partial ξ} = \frac{1}{{Pe}_{k}} \cdot ε^{2} \cdot \frac{\partial^{2} C_{k}}{\partial ξ^{2}} + \frac{1}{{Pe}_{k}} \cdot \frac{\partial^{2} C_{k}}{\partial ζ_{k}^{2}}, k = 1, 2, 3,$ ${\frac{\partial C_{1}}{\partial ζ_{1}}|}_{ζ_{1} = 0} = 0, {\frac{\partial C_{3}}{\partial ζ_{3}}|}_{ζ_{3} = 1} = 0,$ ${\frac{1}{{Pe}_{1}} \cdot \frac{\partial C_{1}}{\partial ζ_{1}}|}_{ζ_{1} = r} = {\frac{1}{{Pe}_{2}} \cdot \frac{\partial C_{2}}{\partial ζ_{2}}|}_{ζ_{2} = r},$ ${\frac{1}{{Pe}_{2}} \cdot \frac{\partial C_{2}}{\partial ζ_{2}}|}_{ζ_{2} = θ} = {\frac{1}{{Pe}_{3}} \cdot \frac{\partial C_{3}}{\partial ζ_{3}}|}_{ζ_{3} = θ},$ ${C_{1}|}_{ζ_{1} = r} = {C_{2}|}_{ζ_{2} = r},$ ${C_{2}|}_{ζ_{2} = θ} = {C_{3}|}_{ζ_{3} = θ},$ where ${Pe}_{k} = \frac{〈u〉 W}{κ_{k} (λ_{k} + \frac{K_{k}}{ϕ_{k}})}, k = 1, 2, 3$ are the Péclet number in zone 1 to 3,

Enhancement of environmental dispersion

To examine the enhancement of the dispersivity, we define $D = D_{E} - 1$ . The values of D with $Pe, α, M, N, r (θ)$ and $ϕ$ are plotted in Fig. 4, which can be compared with the results for the three-layer wetland flow obtained by Wang et al. [33] using the method of concentration moments. Notice that the analyzed results for the asymptotic values of dispersivity in the previous work used different definitions for $ψ_{k}$ and Pe, which also leads to some slight differences in the results.

The values of D are plotted

Comparative example

Regarding to the equally typical three-zone wetland and three-layer wetland, it is interesting to perform a comparative analysis in terms of environmental dispersion.

For the contaminant cloud defined by a given upper limit of concentration $C_{0}$ , according to Eq. (89), the critical length is given by $S = 2 \sqrt{- 4 D_{E} t^{*} \ln [\frac{C_{0} W^{2} \sqrt{4 π D_{E} t^{*}}}{Q (\frac{r}{ϕ_{1}} + \frac{θ - r}{ϕ_{2}} + \frac{1 - θ}{ϕ_{3}})}]},$ which can be rewritten in dimensional coordinates as: $L_{c} = 2 W \sqrt{- 4 D_{E} \frac{t \cdot 〈κ (λ + \frac{K}{ϕ})〉}{W^{2}} \ln \frac{C_{0} W \sqrt{4 π D_{E} \cdot t \cdot 〈κ (λ + \frac{K}{ϕ})〉}}{Q (\frac{r}{ϕ_{1}} + \frac{θ - r}{ϕ_{2}} + \frac{1 - θ}{ϕ_{3}})}} .$

Consider a wetland flow with typical parameters of $ϕ_{1} = 0.9, ϕ$

Conclusions

The renowned Taylor dispersion refers to the longitudinal evolution of the mean concentration over the cross section after an initial stage of the solute. In the presence of vegetation, the mechanism of Taylor dispersion at the micro-scale becomes more complex, inevitably resulting in varied apparent environmental dispersion at the environmental macro-scale. Nepf [7] described the characteristics of velocity and turbulence in flows with aquatic vegetation. Presented in this paper is a

Acknowledgements

This work is supported by the State Key Program for Basic Research (973 Program, No. 2013CB430402), the National Natural Science Foundation of China (No. 11272012) and the Specialized Research Fund for the Doctoral Program of Higher Education of China (No. 20120001110077).

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Transport in a three-zone wetland: Flow velocity profile and environmental dispersion

Highlights

Abstract

Introduction

Section snippets

Formulation for momentum transport and velocity profile

Formulation for concentration transport

Multi-scale analysis for concentration dispersion

Enhancement of environmental dispersion

Comparative example

Conclusions

Acknowledgements

Ecol Econ

Int J Eng Sci

Commun Nonlinear Sci Numer Simul

J Hydrol

Commun Nonlinear Sci Numer Simul

Water Res

J Hydrol

Ecol Modell

Ecol Modell

Commun Nonlinear Sci Numer Simul

Commun Nonlinear Sci Numer Simul

Ecol Modell

Commun Nonlinear Sci Numer Simul

Ecol Eng

Int J Heat Fluid Flow

Int J Heat Mass Transfer

Adv Appl Mech

Int J Eng Sci

Fluid Dyn Res

Int J Eng Sci

The value of the world’s ecosystem services and natural capital

Nature

Wetlands

Flow and transport in regions with aquatic vegetation

Annu Rev Fluid Mech

Indicators for environmental dispersion in a three-layer wetland: extension of Taylor’s classical analysis

Eco Indic

Dispersion of soluble matter in solvent flowing slowly through a tube

Proc R Soc London Ser A Math Phys Sci

The dispersion of marked fluid in turbulent shear flow

J Fluid Mech