Universality of Turbulent Dispersion in a Steady Flow in an Open Channel

We construct a low-dimensional depth-averaged model of turbulent dispersion of tracers in an open channel flow. Two types of the average velocity profile are considered. One is the classical logarithmic profile, and the other is the power profile according to a different similarity hypothesis of the flow with respect to the non-dimensional distance from the wall. Our approach (a) combines the firm justification of the depth-averaging procedure by centre manifold theory and (b) embraces the essential connection between the velocity shear and turbulent diffusion. We deduce an advection-diffusion-dispersion equation for the depth-average concentration for each velocity profile. The equation represents a leading approximation of the transport equation for the tracer and is extendable to include higher-order spatial derivatives. The results for the two profiles are compared in the limit of very large Reynolds numbers.