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Hi, Can you help me out with the hydro dynamic part of the model? |
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Introduction
We propose a unified framework for the mathematical modeling and numerical simulation of the pollution of the Niger River in Bamako in order to contribute to a better understanding of the dispersion pattern of water pollutants. In terms of specific contribution to the current literature, this paper provides the scientific community interested in water pollution issues with a non-trivial application based on new robust algorithms and advanced numerical tools. The model we are interested in consists of a coupling of the shallow-water equations, which govern unsteady water flows in rivers, with an advection-diffusion equation for pollutant transport in water flows.
Mathematical model
The water flows in Niger river is governed by the two-dimensional shallow- water equations [41], often known as Saint-Venant equations. This hydrodynamic model, fairly well known in the literature, is derived from the integration of the three-dimensional incompressible Navier–Stokes equations over the fluid depth under the assumptions of hydrostatic pressure distribu- tion and the uniform velocity distribution along the vertical direction. This system of equations [42] is a suited model widely used for describing the flows in environments where the fluid depth is much smaller than the horizontal scale of motion, which is absolutely the case of the context of this study. On the other hand, for modeling the dynamics of pollutants in water flows, we consider a two-dimensional advection-diffusion equation [43]. Thus, the global model we consider for fully describing the transport of pollutants by shallow-water flows, regarding concurrently flow and transport phenomena, results in the coupling [44] of the shallow-water equations with the advection-diffusion equation. The resulting system of partial differential equations (PDEs) is written in the so-called conservative form as follows:
where
The first equation of the system above represents the mass conservation of the water. It is also referred as continuity equation. The second equation refers to the momentum conservation, while the third equation represents the transport of pollutants by the water flows. The pollutant is supposed to be passive (non-reactive) and we assume that it does not induce any feedback to the water flow.
The bottom shear stress$\boldsymbol{\tau}_b$ is computed using the Manning law as follows:
where$\rho_w$ is the water density, $n_b$ is the Manning roughness coefficient and $\left\Vert\mathrm{\mathbf{q}}\right\Vert$ is the Euclidean norm of the flow discharge. The surface stress $\boldsymbol{\tau}_s$ is expressed as a quadratic function of the wind velocity $\mathrm{\mathbf{w}}$ as follows:
where$c_s$ is the friction coefficient of the wind. It is usually defined by:
where$\rho_a$ is the air density. The additional body forces $\mathrm{\mathbf{f}}$ include atmospheric$f_c$ is defined by:
pressure gradient and tidal potential forces. They will not be considered in this work.
The Coriolis parameter
where$\omega=7.2921\mbox{E}-5~\mbox{rads}^{-1}$ is the angular speed of Earth rotation and $\varphi$ is the geographic latitude of the site.
The diffusion tensor$\mathrm{\mathbf{D}}$ is a $2\times 2$ matrix defined by:
where$D_{xx}$ , $D_{xy}$ , $D_{yx}$ and $D_{yy}$ correspond to the diffusion coefficients of the pollutant in each combination of $x$ and $y$ directions. The diagonal elements $D_{xx}$ and $D_{yy}$ represent the diffusion coefficients along the principal directions and the off-diagonal terms $D_{xy}$ and $D_{yx}$ reflect the correlation between diffusion coefficients along the respective pair of principal directions. In general, the components of the diffusion tensor $\mathrm{\mathbf{D}}$ , accounting for molecular diffusion and turbulent mixing, depend on the flow characteristics and the nature of pollutant considered, including water depth, flow velocity, bottom roughness, wind and vertical turbulence. Since turbulent mixing is the dominant component of dispersion in river, it is feasible to neglect molecular diffusion, which is small scale process that occurs much more slowly than turbulent mixing. Thus, in longitudinal (or streamwise) and transverse reference system, the diffusion tensor can be expressed in the form of a diagonal matrix as follows:
where$D_L$ and $D_T$ are longitudinal (along the flow) and transverse (perpendicular to the flow) dispersion coefficients, respectively. These coefficients are estimated by:
where$u^{\ast}$ represents the shear velocity and $\alpha_L$ and $\alpha_T$ are dimensionless constants quantifying the magnitude of longitudinal and transverse dispersion, respectively. Typical values of these constants for river flows are $\alpha_L=5.93$ and $\alpha_T=0.15$ .
In order to compute the components of the diffusion tensor$\mathrm{\mathbf{D}}$ in the Cartesian coordinate system, the following expressions hold:
where$\theta=\arctan(q_y/q_x)$ represents the angle between the flow direction and the x-axis.
The fields and physical parameters of the model are listed in Table below:
Numerical Results
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