Numerical Solution to a Dispersion Equation

A semi-implicit numerical solution to the differential equation of dispersion in a two-dimensional open channel is developed. Some of the details of the iterative technique and the computer solution are shown. The results for turbulent flows show that the longitudinal distribution of solute concentration approaches a Gaussian (normal) pattern as the dispersion time progresses, and the dispersion coefficient depends jointly on the convective transport and on the turbulent diffusivity. These results provide a rational basis for evaluating the pattern of concentration in the transient phases between the insertion of a slug of solute and its ultimate normal pattern. The substantial computing requirement removes the technique from the realm of practical solutions; it is instead an attempt to explain a heretofore unclear hydraulic phenomenon.