On the exact analytical solution for the spatial moments of the cross‐sectional average concentration in open channel flows

Abstract

This paper shows how an exact analytical solution for the transient‐state spatial moments of the cross‐sectional average tracer concentration in large open channel flows can be derived from the depth‐averaged advection‐diffusion equation resorting to the method of Green's functions, without any simplifying assumption about the regularity of the actual concentration field, the smallness of the fluctuations, or the large space‐time scale of variation of the average concentration gradient (justifying the a priori localization of the problem), which were the basis of the classic Taylor dispersion theory. The results reveal that in agreement with the findings by Aris (1956) and later by others for flows within a conduit, there are an initial centroid displacement and a variance deficit dependent on the specific position and dimension of the initial injection. The second central moment asymptotically tends to the linearly increasing function predictable on the basis of Taylor's classic theory, and the skewness, which is constantly zero for the cross‐sectionally uniform injection, in the case of nonuniform initial distributions tends to slowly vanish after having reached a maximum. Thus, the persistent asymmetry exhibited by the field concentration data, as well as the retardations and the accelerations in the peak trajectory, can be justified without making any a priori assumption about the physical mechanism underlying their appearance, like transient storage phenomena, just by rigorously solving the governing equation for the cross‐sectional average concentration in the presence of nonuniform, asymmetrically located solute injections.