Yongcan Chen , Zhaowei, Liu and Chuang Li

Department of Hydraulic Engineering, Tsinghua University, Beijing 100084, China

Abstract: A non-horizontal multi-layer finite element model has been developed for studying the impact of Three George Project on pollutant transport patterns and water quality in Wanzhou Reach. In this method, the interfaces between layers are non-horizontal, which indicates both the water surface gradient and the variety of topography of riverbed. This method can improve the ability of the model boundary adaptation with actual natural geometry and the water surface. The field observation data is incorporated to determine the flow and pollutant transport parameters and then to test the rationality of these parameters. The good correlation revealed in the comparison between the computed and observed results in Wanzhou Reach indicates that the present model can be served as an effective tool for hydrodynamics and pollutant transport studies in large rivers.

With the Three Gorge Project being built up, it is known that the flow will slow down, the water elevation, topography of the section in the reservoir will be changed remarkably, all of these may have significant impacts on the existing pollutant transport patterns and water quality. For planning and monitoring purpose, it is essential to have a good understanding of the physical and environmental impacts of the Three George Project.

In the last few decades many sophisticated mathematical models for predicting hydrodynamic environment of actual river have been developed. For example, three dimensional finite difference turbulence models, large eddy models and three dimensional finite element models etc., one of these properly used models is the depth-averaged two-dimensional finite differencing schemes owing to its less intensive demand in both formulation setup and computation. This type of scheme is adequate in most pollutant transport studies. However, the depth-average assumption of vertical uniformity cannot accurately represent the pollutant boundary layer variations in the water column. Also finite-difference grids cannot well fit to irregular geometry of the river and this may cause significant discrepancy between the actual situation and the numerical results. To resolve the shortcomings, a three-dimensional non-horizontal multi-layer finite element numerical model has been developed for pollutant transport simulations in large river. Quadrilateral finite elements are employed in the model to construct the horizontal domain to improve the boundary adaptation ability for complicate river configurations. The non-uniformity of the flow and concentration field in the water column is handled by a multi-layer setting in the vertical coordinate. This paper presents the computed results of hydrodynamics and concentration of pollutant in Wanzhou reach of Yangtze River.

In some rivers, the vertical momentum is generally governed by the hydrostatic variation as the vertical velocity is much smaller than the horizontal ones. Thus the hydrodynamic equations and the pollutant transport equation are shown as follows:

in which are the velocities of x, y, z-components in Cartesian coordinates, r is the density of water, P is the pressure, are the Coriolis forces, are shear stresses, which can be written as follows，

The water column is divided into a number of layers and the gradient of variables within each layer are ignored in the present model, which is similar to other multi-layer models. But the interfaces between layers are non-horizontal, which indicates both the water surface gradient and the variety of topography under the water.

Integrating the variables of each layer in the vertical (z) direction, the pollutant concentration and discharges per unit width in the x- and y- direction in the k-th layer are written respectively as

， ，

Thus, the layer-integrated form of the momentum equation in the x-coordinate and the pollutant transport equation in the k-th layer expressed as Eqs.(6) and Eqs.(7), respectively.

x-momentum equation of the k-th layer:

(6)

Pollutant transport equation of the k-th layer:

(7)

Here ，denotes the net pollutant flux through the interface.

With the impermeable assumption of the river bed, the interfacial vertical velocity can be induced from the continuity equation of fluid mass after layer-integrated, which is shown as

(8)

As the surface vertical velocity varied in time with the free surface elevation, the free surface elevation equation is described as

(9)

in which L is the total number of the layers.

The existence, uniqueness and accuracy of the solutions of the above equations are heavily dependent on the prescribed boundary conditions. On the open boundary, the velocities and pollutant concentration are specified at the upstream water interface at each layer of the system and the water elevation is prescribed at the downstream surface. On the close boundary, the stresses and the zero gradient condition of concentration are specified.

(10)

To improve the boundary adaptation ability for complicate natural river configuration, four sided isoparametric finite elements are employed to discretized the horizontal domain. Velocities, water elevation, pollutant concentration, and the terms associated with these variables for the k-th layer in each element are interpolated based on the isoparametric interpolation function. Some of these variables are shown as :

(11)

Applying the vertical layer-integration and the standard Galerkin finite element procedure, the hydrodynamic and pollutant equations for the k-th layer are expressed as follow.

in which A, P, Q, R, S, y are the coefficient matrices in the whole simulation field.

A special kind of upwind finite element scheme suggested by T. J. R. Hughes is used to solve the numerical oscillations introduced by the strong convection term in governing equations. In this method the numerical quadrature rule for the convection term is modified to achieve the upwind effect, meanwhile all other terms of the Galerkin formulation remain unaltered, The unsymmetrical location of the evaluation point controls the degree of upwind. This formulation of upwind finite elements is extremely simple to implement into existing computer codes employing Galerkin finite element methods for convective transport phenomena.

To examine the three-dimensional model adaptability to real natural river settings and to verify the accuracy of simulated results, the model was employed to predict the flow and concentration field near the side discharge outlet of the Fuling Phosphate Fertilizer Factory in a dry season, January 1998. The model is applied to the simulation of flow and pollutant transportation in the reach of Fuling, Yangtze River. Through comparing the computed results with the field observations, which are measured by the Hydrology Bureau of the Water Conservancy Committee of Yangtze River, the other related computational parameters such as horizontal eddy coefficient etc. are determined. These determined parameters are shown in Table1. Fig. 2 shows the comparison of the observed velocity and concentration profiles and the computed profiles respectively. The predicted values agree rather well with the observed values at the given sections.
             Table 1 Parameters for the simulation

In this study, the model area in Wanzhou Reach of Yangtze River is from Tuokou to Saiwangba, which is12 km long and 400m wide. The outlet of the wastewater discharge locates at Zhuxihe, which is shown in Figure 3. The water elevations at Tuokou and Saiwangba is 100.74m and 97.28m, there are 3.46m with the difference between these two sections. The runoff of the river is 4120m³/s in this simulation. The discharge of wastewater is 54100m³/d, and the concentration of CODcr is 548mg/l. The flow field of the modeled area is horizontally divided into a number of quadrilateral elements by using a grid generation model based on the boundary-fitted coordinate system. The vertical water column is divided into eight layers of horizontal planes. The number of layers used in the water column is dependent on the water depth, and the number of elements used in each layer is dependent on the area of each horizontal plane. The criteria of adopting number also include the desired accuracy, computer capability and economic feasibility. Near the outlet of the discharge the number of element increases to keep the simulation accuracy, meanwhile in the part far away from the outlet the number of element decreases to reduce the computation time and storage.

Fig.3(a) and Fig.3(b) show the velocity field at the water surface and the middle layer for the modeling area, it is found that the velocity vectors are changed with the bank configuration and the water depths. The good agreement between the computed velocity and observed data at the downstream section(Saiwangba) is shown as Fig.4. The distribution of COD_mn concentration is shown as Fig.5. According to the Surface Water Standard of P.R. China, there are about 410m long and 35m wide for the pollution zone in which the COD_mn concentration is larger than the Class II (COD_mn=4ml/g) of the Standard due to the wastewater discharge. After the Three Gorge Project is completed, the water elevation is increase to 175m above the sea level, so the velocity of the river is much smaller than this simulation, it is clear that the length and width of pollution zone should be increased if the same discharge of wastewater.

A 3-D Multi-layer Finite element model has been developed for studying the impact of Three George Project on water quality along Yangtze River. The non-uniformity of flow and pollutant distributions in the water-column are represented by multiple vertically averaged layers. Then the complicated 3-D Flow is simplified as several vertically coupled 2-D problems.

The model is applied to the reach of Wanzhou,Yangtze River to simulate the flow and pollutant transport patterns during a dry season. Comparison of the computed fields are made against the corresponding observed values. The numerical results indicate that this model can provide a realistic and proper framework for pollutant transport patterns predictions in actual rivers.

[1]  Wai,O., Chen, Y., Lu, Q., Multi-layer simulation of sediment transport around a breakwater, Proc. 27th Congress of Inter. Association of Hydraulic Research, IAHR, Part B-1, San Francisco, USA,725-730,1997.

[2]  Li Y. S. and Zhan J. M., An Efficient Three-Dimensional Semi-Implicit Finite Element Scheme For Simulation of Free Surface Flows. Int. J. Numer. Meth. Fluids, Vol. 16, pp. 187-198 1993.

[3]  Hughes J. R., A Simple Scheme For Developing “Upwind” Finite Elements. Int. J. Numer. Meth. Engr., Vol. 12, pp.1359-1365, 1978.

[4]  Chen Yongcan, Liu Zhaowei, Li Chuang, The Investigation and Numerical Simulation of Pollution Zone for The Wastewater Discharge from Fuling Phosphate Fertilizer Factory in Three Gorge Reservior, Proc. of 2000 China-Japan Joint Symposium on Green Science and Technology, USTC Press , 2000.