Pollution

Resolving the transport equation to visualise the evolution of a pollutant cloud

About

At the Ecole Centrale de Lyon, we had to elaborate a code to simulate the propagation of a fictional pollutant cloud, and analyze the concentration of pollutant to see if it was noxious.

How it works

Modelisation of the problem

3 factories are emitting pollutant, we have access to the wind velocity at $\Delta t$ intervals. The propagation of the pollutant cloud is modelled by the transport equation.

We are modelling the equation using :

$$ \frac{\partial \overline{C}}{\partial t} + \overline{u}\cdot \nabla \overline{C} = D_t\Delta\overline{C} + S + F$$

where $\overline{C}$ is the averaged concentration of pollutant, $D_t$ is the diffusion coefficient, $S$ is the source term, it represent the 3 factories that emit pollutants, and F represent the deposition (dry or wet) of the pollutant from the cloud in the soil.

The deposition of pollutant is modelled by the equation

$$ F = k\cdot \overline{C}$$

Numerical approximation

The laplacian is approximated using the 5 point method :

$$\Delta u(x,y) = \frac{4 U_{i,j} - U{i+1, j}- U{i-1, j}- U{i, j+1}- U{i, j-1}}{h^2}$$

where h represent the spatial step.

Likewise, $$\frac{\partial \overline{C}}{\partial t} = \frac{C_{i,j}^{(n+1)} - C_{i,j}^{(n)}}{\Delta t}$$

Results

Several figures and data on the deposited pollutants were obtained, leading to the conclusion that the factories were noxious.

Workspace description

The main.m program executes the code and shows the results. The core of the simulation is handled in the simulation.m. The others files are mainly utility functions.