Analytical study of turbulent pollutant dispersion near a low hill

An analytical methodology is developed to study the pollutant dispersion in a turbulent wind flow over a two-dimensional hill with a small slope. As in a typical boundary layer problem, the flow domain is divided into an inner and an outer region: the inviscid outer region is further subdivided into an upper and a middle layer while the viscous inner region is subdivided into a shear stress and an inner surface layer. Based on the Reynolds-averaged Navier-Stokes equations and the continuity equations, closed form analytical solutions of the stream functions and velocities are readily obtained for all regions in the domain. The velocity information is then imported into the diffusion equation, and the pollutant concentration distribution is readily solved. For reasons of turbulent shear, a variational method with adjustments to the streamline coordinate system is used to obtain an accurate solution of the pollutant concentration. Results show that when the source is located in the upper layer, the concentrations decrease with distance along the upwind side of the hill and tend to reach a constant value rapidly near the hilltop. Similar results are observed when the source is located in the middle layer. However, due to the reduction of wind speed in the middle layer, the concentrations become saturated at a later upslope position as compared to the source in the upper layer. This methodology is shown to be able to provide a quick and accurate estimate of local pollutant patterns and can be applied to any flow field provided that the streamlines can be specified through the velocities.