Turbulent dispersion from sources near two-dimensional obstacles

Abstract

The first part of the paper is an extension of the statistical theory of turbulent diffusion to the problem of turbulent diffusion from a point or line source in an inhomogeneous straining flow such as flow round a two-dimensional body. The new effects which have to be considered are the convergence and divergence of mean streamlines, the inhomogeneity of the mean and turbulent velocities, and the presence of a boundary. We assume that the upstream turbulence intensity$u^{\prime}_{\infty}/\overline{u}_{\infty} $is weak, i.e.$u^{\prime}_{\infty}/\overline{u}_{\infty} $[Lt ] 1, and that molecular diffusion is negligible, i.e.$(\overline{u}_{\infty}/u^{\prime}_{\infty})^2[D/(a\overline{u}_{\infty})] $[Lt ] 1,Dandabeing the molecular diffusivity and the scale of the obstacle respectively. The theory predicts the mean-square dispersion of the plume about the mean streamline through the source in terms of the Lagrangian statistics of the turbulence. Making the further assumption that the scaleLEof the turbulence is large enough to satisfy the condition that ($(u^{\prime}_{\infty}/\overline{u}_{\infty}) $) (a/LE) [Lt ] 1, it is shown that the turbulent dispersion can be calculated in terms of the Eulerian statistics, which can either be measured or in some cases calculated. In the second part we analyse diffusion from various sources in potential flows over two-dimensional obstacles assuming constant (or variable) eddy diffusivity, and compare the results with those of the more rigorous statistical analysis for sources one or two diameters upwind of the obstacle. However, unlike the statistical analysis, this eddy-diffusivity analysis can also be extended to calculate diffusion from sources placed some distance upwind of an obstacle, and an example is given of how this analysis may be applied to calculating concentrations on hills due to distant sources.