Analytical solutions for contaminant transport in open channel flows and underlying slow zones

A general analytical model, with three layers, based on the method of moments, is developed for studying the flow and solute transport between the open channel (fast zone) and two slow moving zones below. The transport in each domain is described by a two-dimensional advection-diffusion equation with the exchange between the three domains modelled through boundary conditions that ensure the continuity of concentration and flux at the interface. An exponential decreasing velocity profile of the middle zone links the velocity profile of the open channel at the top and the velocity profile of the bottom layer which is almost stagnant. Analytical expressions for the zeroth, first and second spatial moments of concentration are derived using the Aris' method of moments and exact expressions for the mean solute velocity and effective dispersion coefficient are also derived. Flow and transport in the open channel and its adjacent zone are diffusion and advection driven, but in the case of the bottom zone the process is more dominated by diffusion. Exact analytical expressions derived for mean solute velocity and dispersion coefficient make this study very valuable for further applications in a relevant context. An important application of the study is its direct relevance to an upcoming research on dispersion in flows with submerged vegetation where it is highly recommended that the submerged vegetation zone be divided into two zones. It is seen that the mean solute velocity increases with an increase in friction velocity and it is also sensitive towards the ratio of the slow zone depth over the depth of the slower zone. The results show that the effective dispersion coefficient decreases with increase in relative diffusivity. The combined effect of increased diffusivity and friction velocity is to bring down the value of the effective diffusion coefficient. The model is verified using the results for limiting cases whose analytical solutions are known. Since the model parameters can be measured independently, the analytical solutions developed in this study can be used as a practical tool to model water quality in rivers and underlying sublayers.