Problems in turbulent dispersion

Abstract

Solutions to a number of turbulent dispersion problems, involving a contaminant issuing from a steady source into uniform, steady open channel flow, are presented. These problems include the two and three-dimensional dispersion of a neutrally buoyant contaminant and the two-dimensional dispersion of buoyant particles and are modelled with the diffusion equation incorporating turbulent diffusion coefficients. In order to ensure that the solutions simulate the physical processes as accurately as possible the experimentally determined logarithmic velocity profile and the theoretically deduced parabolic diffusivity are used. The solutions take the form of one or two eigenfunction expansions, the eigenfunctions and eigenvalues of which are governed by Sturm-Liouville theory. Generally the power series method for solving ordinary differential equations is employed to derive the eigenfunctions and eigenvalues and in nearly all cases this method is found to be accurate, straightforward in its use and efficient with computing resources. Much useful information is deduced from the eigenfunctions and eigenvalues. The rate at which equilibrium conditions are approached and the ideal source position, that from which the contaminant is most rapidly mixed, come naturally from these quantities. An experimental programme, with the aims of verifying the theoretical solution for vertical dispersion of a neutrally buoyant contaminant and measuring the lateral turbulent diffusion coefficient, is described. The two-dimensional results for vertical mixing strongly support the theoretical predictions, using the measured logarithmic velocity profile and the deduced parabolic diffusivity in the turbulent diffusion equation, and indeed confirm the location of the ideal source derived from theory. The values of the depth-averaged lateral diffusivity obtained from the experiments lie at the lower end of the range of values obtained by other experimentalists. A reanalysis of these previously published results demonstrates that, provided the natural turbulence of a wide channel is the only mixing mechanism present, the depth-averaged lateral diffusivity, non-dimensionalised by the flow depth and shear velocity, is in fact independent of all flow parameters, except when the friction factor is small. The dependence of the rate of lateral spreading on height in the flow and the location of the source demonstrates, at least qualitatively, that the vertical dependence of the lateral diffusivity is in essence the same as the velocity distribution. Verification of the theoretical solution for dispersion of buoyant particles is achieved with the experimental results of Jobson and Sayre (1970) which indicate that the theoretical model is valid for fine particles dispersing in strongly turbulent flow.