Analytical solution for scalar transport in open channel flow: Slow-decaying transient effect

Abstract

Summary It is well known that the extensively applied Taylor dispersion model only predicts the longitudinally distributed mean concentration. While at the same time, applications as the risk assessment for toxic pollutant transport in environmental fluid flows require detailed information on the cross-sectional concentration distribution. As shown by some recent progress (Wu, Z., Chen, G.Q., 2014, J. Fluid Mech., 740, 196–213.), the deviation of transverse concentration from the mean can be remarkable for a very long time, which is termed as the slow-decaying transient effect. Thus it is important to examine the process of concentration evolution for scalar transport in laminar open channel flow. In this paper, the idealized case of a uniform and instantaneous scalar release across the channel is analytically explored by a two-scale perturbation analysis. The validity of the Taylor dispersion model for the mean concentration is discussed by the obtained analytical solution. For the first time, the two-dimensional concentration distribution for the open channel flow is explored analytically. Corresponding time scales for the concentration evolution are determined, indicating that the process for the vertical concentration difference to diminish will be much slower than that for the mean concentration to become Gaussian. Dominated by the so-called slow-decaying transient effect, the uniform vertical distribution needs to be modified to predict the vertical concentration distribution correctly.