The Accurate Dynamic Modelling of Contaminant Dispersion in Channels

Throw some dye into a flowing river; how is the dye going to disperse along the river? The dynamics may be modelled by Taylor’s one-dimensional model of shear dispersion. We discuss the creation and utility of models of the dispersion of the dye, using invariant manifold theory. The models are effective exponentially quickly. We describe a one-mode centre manifold and a two-mode and a three-mode invariant manifold model. From the analysis, we decide under what conditions these three models are valid. Using the geometric picture provided by the theory of invariant manifolds, we find a complete set of initial conditions for the models so that the evolution of the full and approximate systems are identical after a short time. Throughout, a comparison of the three models is made to identify the practical situations when each model will be preferred.