Predictability of tracer dilution in large open channel flows: Analytical solution for the coefficient of variation of the depth‐averaged concentration

A large‐time analytical solution is proposed for the spatial variance and coefficient of variation of the depth‐averaged concentration due to instantaneous, cross sectionally uniform solute sources in pseudorectangular open channel flows. The mathematical approach is based on the use of the Green functions and on the Fourier decomposition of the depth‐averaged velocities, coupled with the method of the images. The variance spatial trend is characterized by a minimum at the center of the mass and two mobile, decaying symmetrical peaks which, at very large times, are located at the inflexion points of the average Gaussian distribution. The coefficient of variation, which provides an estimate of the expected percentage deviation of the depth‐averaged point concentrations about the section‐average, exhibits a minimum at the center which decays like t −1 and only depends on the river diffusive time scale. The defect of cross‐sectional mixing quickly increases with the distance from the center, and almost linearly at large times. Accurate numerical Lagrangian simulations were performed to validate the analytical results in preasymptotic and asymptotic conditions, referring to a particularly representative sample case for which cross‐sectional depth and velocity measurements were known from a field survey. In addition, in order to discuss the practical usefulness of computing large‐time concentration spatial moments in river flows, and resorting to directly measured input data, the order of magnitude of section‐averaged concentrations and corresponding coefficients of variation was estimated in field conditions and for hypothetical contamination scenarios, considering a unit normalized mass impulsively injected across the transverse section of 81 U.S. rivers.