On the re-aeration coefficient in channels of complex shape

The recovery of rivers from spills of organic effluents is influenced by the diffusion of oxygen from the atmosphere, which is quantified by the re-aeration coefficient. Many tens of formulae for estimating this coefficient from simple hydraulic variables exist. As well as it being difficult to choose the most appropriate formula for any particular river reach, another issue exists for rivers of complex cross-sectional shape. In these cases there are significant transverse variations in water depth and flow velocity. Hence, the re-aeration coefficient must vary transversely also. The paper presents initial results from a theoretical analysis aimed at exposing the significance for estimated re-aeration coefficients of properly capturing the transverse heterogeneity of the physical processes. Three strategies for estimating the reaeration coefficient for a channel of complex shape, consisting of a rectangular main channel surrounded by two symmetrical rectangular floodplains, were considered. Firstly, a simplistic approach in which the coefficient was evaluated only for the hydraulic conditions in the main channel, and expected to be dubious because it ignored transverse variations in the hydraulic conditions. Secondly, a naive approach in which the coefficient was evaluated using cross-sectional average hydraulic conditions, and expected to be better than the simplistic approach because it attempted to recognize transverse variations in the hydraulic conditions. Thirdly, a robust approach in which the coefficient was evaluated as the cross-sectional average of three local values of the coefficient (one value for each flow zone, based on local hydraulic conditions), and expected to give the most reliable results because the transverse heterogeneity of the hydraulic conditions was properly captured. Using a typical empirical formula for the re-aeration coefficient and a modified flow resistance formula, general expressions for the re-aeration coefficient for each strategy were obtained in terms of the ratios of flood plain roughness to main channel roughness (gamma), flood plain width to main channel width (beta) and flood plain water depth to main channel water depth (eta). Computations were undertaken for 1 <gamma <4, 0.5 <beta <4 and 0.05 <eta <0.4. The results show that in comparison to the robust approach the simplistic approach overestimates the coefficient by up to 100%, with their ratio increasing with increasing gamma and beta, but gradually decreasing with increasing eta. The results for the naive approach are more complex. In comparison to the robust approach: when gamma is low, it overestimates the coefficient (by up to 10%) and beta has little effect, but when gamma is high, it underestimates the coefficient (by up to 15%) and their ratio increases towards unity with increasing beta; also their ratio gradually decreases with increasing eta for all gamma and beta. In conclusion, although it ay be tempting to evaluate the re-aeration coefficient from cross-sectional average hydraulic conditions, significant errors may be incurred.