The governing equations of continuity and momentum of the three-dimensional unsteady hydrodynamic model can be written as follows:

where u_i (i = 1, 2, 3) are the Reynolds-averaged flow velocities (u, v, w) in Cartesian coordinate system (x, y, z); t is the time; r is the water density; p is the pressure; n is the fluid kinematic viscosity; is the Reynolds stress; and f_i are the body force terms.

The turbulence Reynolds stresses in Equation (2) are approximated according to the Bousinesq’s assumption. These stresses are related to the mean velocity gradients and the eddy viscosity coefficients. The horizontal eddy viscosity coefficient is computed using the Smagorinsky scheme [12]:

where Dx and Dy are the grid lengths in x and y directions, respectively. The parameter a ranges from 0.01 to 0.5.

For wind-driven flow, the vertical eddy viscosity is mainly determined by the wind stresses. Several formulas were presented to calculate the wind-induced eddy viscosity (Section 2.2).

The free surface elevation (h_s) is computed using the following equation:

where u_s, v_s and w_s are the surface velocities in x, y and z directions; h_s is the water surface elevation.

The pollutant transport equation can be expressed by:

in which u, v, w are the water velocity components in x, y and z directions, respectively; w_s is the settling/floating velocity; C is the pollutant concentration in water column; D_x, D_y and D_z are the diffusion coefficients in x, y and z directions, respectively; is the effective source term.

In a natural lake or the Oakville flood inundation zone, wind stress is the important driving force for flow circulation. For simulating the wind driven flow, the distribution of vertical eddy viscosity is required. A parabolic eddy viscosity distribution was proposed by Tsanis [13] based on the assumption of a double logarithmic velocity profile. The eddy viscosity was expressed as:

in which l is the numerical parameter; z_b and z_s are the characteristic lengths determined at bottom and surface, respectively; u_*s is the surface shear velocity; H is the water depth. To use this formula, three parameters, l, z_b and z_s have to be specified. For some real cases with very small water depths, it may cause some problems using this formula to calculate eddy viscosity.

Koutitas and O’Connor [14] proposed two formulas to calculate the eddy viscosity based on a one-equation turbulence model. Their formulas were:

where

in which, l is the numerical parameter; h_z is the nondimensional elevation,.

In this study, a new formula was proposed to calculate the wind-induced eddy viscosity based on experimental measurements conducted in a laboratory flume with steady-state wind driven flow reported by Koutitas and O’Connor [14]. The vertical distribution of eddy viscosity was measured and used to fit a calculation formula.

In the proposed formula, the form of eddy viscosity was similar to Koutitas and O’Connor’s assumption (Equations (7) and (8)) and expressed as:

Based on measured data, a formula was obtained to calculate the vertical eddy viscosity:

Figure 1 compares the vertical distributions of eddy viscosity obtained from experimental measurements and formulas provided by Tsanis (Equation (6)), Koutitas (Equations (7) and (8)), and the proposed formula (Equation (11)). The proposed formula can be used to calculate the eddy viscosity over the full range of water depth. At water surface (), Equations (6) and (8) show the eddy viscosity is zero or very closed to zero. The newly proposed formula shows due to the effect of wind shear, the eddy viscosity at the water surface is about 16% of the maximum eddy viscosity. This formula was used to calculate the vertical eddy viscosity in this study.

Wind stress is generally the dominant driving force for flow currents in natural lakes. The wind shear stresses at the free surface are expressed by

where is the air density; U_wind and V_wind are the wind velocity components at 10 m elevation in x and y directions, respectively. Although the drag coefficient C_d may vary with wind speed [14,15], for simplicity, many researchers assumed the drag coefficient was a constant on

the order of 10⁻³ [16,17]. In this model, C_d was set to 0.001, and this value is applicable for simulating the wind driven flow in several natural lakes [18,19].

On the free surface, the normal gradient of pollutant concentration was set to be equal to zero. In the vicinity of the solid walls and bed, the gradients of flow properties were steep due to wall effects. The normal gradient of concentration on the wall was set to be zero.

There are many dry areas in the flooded zone, which represent the buildings of farmer facilities and islands, and wet and dry technique was applied to those areas. In the model, non-slip and the zero normal gradients boundary conditions were set for simulating the flow velocity and pollutant concentration, respectively.

The numerical model was developed based on the finite element method. Each element is a hexahedral with three levels of nine-node quadrilaterals, and the governing equations are discretized using a 27-node hexahedral (Figure 2).

Staggered grid is adopted in the model. The grid system in the horizontal plane is a structured conformal mesh generated on the boundary of the computational domain. In vertical direction, either uniform or non-uniform mesh lines are employed. In order to get more accurate results, the non-uniform mesh lines are placed with finer resolution near the wall, bed and free surface.

The unsteady equations are solved using the time marching scheme. A second-order upwinding scheme is adopted to eliminate oscillations due to advection. In this model, a convective interpolation function is used for this purpose. This function is obtained by solving a linear and steady convection-diffusion equation analytically over a one-dimensional local element shown in Figure 2. Although there are several other upwinding schemes, such as the first order upwinding, the second order upwinding and Quick scheme, the convective interpolation function is selected in this model due to its simplicity for the implicit time marching scheme [11].

The velocity correction method is applied to solve the dynamic pressure and enforce mass conservation. Provisional velocities are solved first without the pressure term, and the final solution of the velocity is obtained by correcting the provisional velocities with the pressure solution [11]. The system of the algebraic equations is solved using the Strongly Implicit Procedure (SIP) method.

Flow fields, including water elevation, horizontal and vertical velocity components, and eddy viscosity parameters were computed by CCHE3D. After getting the effective source terms SS, the pollutant concentration distribution can be simulated by solving pollutant transport Equation (5) numerically.

A laboratory test case was carried out in a wind-wave flume by Koutitas and O’Connor [14] to study the winddriven flow. The length and width of the measuring flume were 5 m and 0.2 m, and the water depth was 0.31 m. The vertical current profile near the middle section was measured. The developed numerical model was applied to simulate the velocity profile of the wind driven flow for the experimental case. A non-uniform grid with 21 vertical nodes with fine resolution near surface and bed were used for model validation. In the numerical modeling, both Equations (6) and (11) were used for calculating the eddy viscosity, and the simulated vertical velocity profiles were compared with experimental measurements (Figure 3). Numerical results are generally in good agreement with experimental measurements. Near the water surface, the model underestimated the velocity using Tsanis’s formula (Equation (6)) to calculate the eddy viscosity; while it overestimated the velocity by us-

ing the proposed formula (Equation (11)). Near the bottom, the simulation obtained by using the proposed formula gave better predictions.

The proposed numerical model was tested against an analytical solution for predicting the concentrations of a non-conservative substance in a hypothetical one-dimensional river flow with constant depth and velocity. At the middle of the river, there is a point source with constant concentration (Figure 4). Under this condition, the concentration of the substance throughout the river can be expressed as:

where U is the velocity; C is the substance concentration; D_x is the dispersion coefficient; x is the displacement from point O; and K is the decay rate. An analytical solution given by Thomann and Mueller [20] is:

where C₀ is the concentration at O point. In this test case, it was assumed, the water depth is 10 m and D_x is 30 m²/s, the values of K are 0, 1.0/day and 2.0/day, respectively, Figure 5 shows the concentration distributions obtained by the numerical model and analytical solution. The maximum error between the numerical result and analytical solution is less than 2%.

Due to a long period of heavy precipitation, a 500-year flood occurred in June 2008 and affected portions of the Midwest United States. In the evening of June 14, a levee

along the Iowa River just upstream of Oakville failed, and the water in the river rushed through the broken levee and completely flooded the small town. Figure 6 shows the flood zone and the location of the levee breaching. As the flooding area was governed by river levees and hilly area, an inundation area was formed and the high water level in this zone tented to remain for several weeks. It was reported that the length of the damaged levee was about 1100 m, and it was not completely fixed until July 1 [2,3,21]. It was estimated from the high water mark, the high water level was about 2.5 meter above the ground [22,23]. It was also observed that some oily pollutants were leaked and transported on the water surface [5]. To study the pollutant transport in the inundation zone, this flooding area can be treated as a natural lake. As this area is relatively flat, the wind is the major driving force of the water flow. To demonstrate the capabilities of the developed model, it was applied to simulate the wind-induced flow and pollutant transport.

Figure 7 shows the computational domain and topography obtained from 30 m resolution of USGS Digital Elevation Model (DEM). The inundation zone was an enclosed water body with the north-south axis spanning about 16 km, while the shorter east-west axis was about 8 km. There were many residential houses and farm facilities in this flooding zone, which could affect the flow fields. In the numerical model, they were represented with a number of dry nodes. The computational mesh was generated using the CCHE Mesh Generator. In the horizontal plane, the computational domain was represented by a 199 × 300 irregular quadrilateral mesh. In the vertical direction, it was divided into 8 non-uniform layers and mesh lines conformed to the curved and irregular bed bathymetry. The vertical distribution of the non-uniform grid was generated using the flexible and powerful two-direction stretching function with finer spacing near the surface and bottom [24].