Transverse Mixing in Rivers With Longitudinally Varied Morphology

Abstract

Predicting the spatial distribution of pollutant concentration in rivers downstream from point sources, e.g., industrial or urban water waste releases or accidental spillings, is critical for assessing water quality and aquatic wildlife sustainability. At large scale, a pollutant in a river mostly spreads in the longitudinal direction, but more locally and closer to the source, spreading also takes place in the transverse direction, until a uniform concentration is reached across the river width (Rutherford, 1994). 1D or 2D numerical modeling of pollutant dynamics (Abderrezzak et al., 2015;Seo et al., 2016) then requires values of both longitudinal and transverse mixing coefficients.The transverse mixing coefficient y E (m 2 /s) is a catch-all coefficient lumping the contributions of different processes to the transverse mixing: molecular diffusion, usually negligible, turbulent diffusion, and dispersion due to secondary circulations (

Formulations and Diffusivity Coefficients of the 2D Depth‐Averaged Advection‐Diffusion Models: A Literature Review

“…For meandering open channels, Lee and Kim (2012) and Park et al (2020) justify this assumption by the large curvature of the flow and thus the strong secondary currents. According to Fischer et al (1979), this explains why the longitudinal dispersion coefficient derived theoretically by Elder (1959) is very low compared to mixing coefficients reported by river studies. Very few quantitative validations of this so-called "velocity-based" method (by comparing the resulting dispersion coefficients with the diffusivity-or mixing-coefficients obtained by scalar release, often labeled "concentration" or "routing" method) could be found in the literature.…”

“…Seven studies either neglect or do not report the considered diffusivity values, and six others use arbitrarily selected values. In seven other studies, the diffusivity values are taken from the literature of scalar dispersion in open channels and rivers, that is, mostly from Elder (1959), Fischer (1973), or Fischer et al (1979.…”

“…As such, the determination of tensor D for real-world applications remains particularly challenging. The coefficients D x and D y correspond to gradient-diffusion, also referred to as "Fickian dispersion terms" or "Fickian turbulent diffusion" (Fischer et al, 1979), while the terms involving D xy and D yx are referred to as "cross-dispersion terms" (Fischer, 1978;Lee & Kim, 2012).…”

“…It is interesting to notice the appearance of cross-dispersion terms brought by the depth averaging, even in cases with a local diffusivity tensor ε t strictly diagonal. Indeed, following the works of Taylor (1954) in pipe flows and of Elder (1959) in 1D open-channels, Fischer et al (1979) showed that the second-order z-derivative of the local concentration is proportional to the gradients of the depth-average concentration along x and y axes (see their Equation 4.1.10), leading to Equation 11. Several strategies have been used for evaluating the vertical distribution of the two velocity components necessary for the evaluation of Equation 11.…”

“…As detailed in section d of Table 2, the first method consists in using coefficients taken from published data sets, such as provided by Elder (1959), Fischer (1973), or Fischer et al (1979), among other more recent works as that from Seo et al (2016). Such data are readily available from recent review articles such as Kashefipour and Falconer (2002), Gualtieri and Mucherino (2007), Y.…”