Advection-diffusion in reversing and oscillating flows, II, Flows with multiple reversals

Motivated by the need to understand effluent dispersion in shallow tidal waters, a two-dimensional analysis of advection and diffusion in a reversing flow has been carried out. The flow speed varies linearly with time, passing through zero at time t = 0. A point source discharges contaminant into this flow at a steady rate, so that water which is close. to the source around the time of flow reversal will become highly contaminated. Thus a peak in the contaminant concentration field will appear, moving downstream after the reversal at a speed close to that of the flow. This concentration peak has certain characteristics similar to a cloud of contaminant from an instantaneous discharge at t = 0. The solution of the advection-diffusion equation is in the form of an integral of concentration fields due to instantaneous releases of contaminant at all previous times. At large times after the flow reversal, asymptotic analysis yields good approximations to this integral. The use of Laplace's method is equivalent to ignoring longitudinal diffusion (the boundary-layer approximation); however, the expansion obtained in this way is not uniformly valid near the concentration peak, indicating that longitudinal diffusion plays an important role in the development of this peak. Uniformly valid expansions are obtained for the concentration around the peak, and also around the source where the boundary-layer approximation always breaks down. Numerical integration has also been carried out, the results being used to produce contour plots of concentration for various times either side of the flow reversal. 1