An improved formula for bed-load rate in open channel flows with emergent vegetation

There is an urgent need to predict the bed-load transport rate in vegetated river ecosystems to support restoration efforts. In response, we have developed a novel model for estimating the effective shear stress acting on the riverbed. This model is based on the energy equation and considers the intrinsic relationship between energy loss in the mean flow and turbulence generated by vegetation in open channel flows with emergent vegetation. Using this bed shear stress model, we assessed the performance of the Meyer-Peter-Muller (MPM) formula in predicting the bed-load transport rate in vegetated flows by comparing it with collected literature experimental data. The results revealed that the MPM formula does not provide accurate predictions. It tends to overestimate the bed-load transport rate when the dimensionless effective shear stress is approximately less than one and underestimate them when the dimensionless effective shear stress is approximately greater than one. This suggests that vegetation enhances and decreases the sediment transport rate when the dimensionless effective shear stress is approximately larger or lower than one, respectively. Consequently, we modified the coefficients of the MPM formula using extensive experimental data, leading to the development of a novel predictive formula for the bed-load transport rate in vegetated flows. This new formula outperforms existing literature equations and is effective for predicting the bedload transport rate, even for umbrella-like vegetation.