One Dimensional Contaminant Transport with Turbulence Effect: Modeling and Solutions

This paper presents a one-dimensional model for atmospheric contaminant transport that accounts for advection, diffusion, and turbulence effects. The classical advection-diffusion equation is extended through Reynolds decomposition and hence averaging to incorporate turbulence via eddy diffusivity. While analytical solutions for such models are typically intractable under real-world conditions with complex boundaries, we derive an exact solution for idealized cases with general initial condition and Dirichlet boundary conditions, providing a valuable benchmark for numerical validation. To address more realistic scenarios, numerical solutions are developed using finite difference (FD) and Crank-Nicolson (CN) methods. This study explores the impact of eddy diffusivity and temporal dynamics on pollutant concentrations within idealized scenarios, benchmarking numerical methods against a derived analytical solution. The analysis reveals that the CN method outperforms FD method in accuracy, particularly in diffusion-dominated regimes (Pe<<1), owing to its unconditional stability and second-order temporal precision. By combining analytical and numerical approaches, this study provides a robust framework for simulating turbulent transport where purely analytical solutions are impractical. The findings show that both methods effectively predict pollutant concentrations under turbulent conditions, offering practical insights for environmental impact assessments and pollution mitigation strategies.