Application of a slow-zone model to contaminant dispersion in laminar shear flows

Abstract

A slow-zone model which was developed for dispersion in two-dimensional turbulent open channel flow is here applied to contaminant dispersion in laminar shear flow in a circular pipe or between parallel plates. In this model the flow is divided into two zones—a fast zone in the middle of the flow and a slow zone near the wall. The resulting pair of coupled linear dispersion equations is solved by Fourier transformation and the Fourier integrals are inverted approximately. The significance of the results is that explicit analytic expressions are obtained for the fast zone, slow zone and cross-sectional average contaminant concentrations. These expressions are valid for all longitudinal positions and all time following contaminant discharge. The results show that laminar dispersion can be described in two stages. In the first stage, corresponding to small times after contaminant discharge, the mean contaminant distribution ( c) consists of a leading Gaussian (mostly in the fast zone), followed by a trailing Gaussian (mostly in the slow zone), and the peak values decay exponentially with time. In the second stage (larger times) c gradually approaches a single Gaussian which travels with the mean speed in accordance with the Taylor model. These theoretical results are shown to be in reasonable agreement with previously calculated exact (numerical) solutions for laminar pipe or channel flows.