Feedback between bed load transport and flow resistance in gravel and cobble bed rivers

To calculate bed load, engineers often use flow resistance equations that provide estimates of bed shear stress. In these equations, on the basis of the estimate of the appropriate hydraulic radius associated with the bed only, the bed roughness k(s) is commonly set as a constant, whatever the bed load intensity. However, several studies have confirmed the existence of feedback mechanisms between flow resistance and bed load, suggesting that a flow-dependent bed roughness should be used. Therefore, using a data set composed of 2282 flume and field experimental values, this study investigated the importance of these feedback effects. New flow resistance equations were proposed for three flow domains: domain 1 corresponds to no bed load and a constant bed roughness k(s) = D (where D is a representative grain diameter), whereas domain 3 corresponds to a high bed load transport rate over a flat bed with a constant bed roughness k(s) = 2.6D. Between these two domains, a transitional domain 2 was identified, for which the bed roughness evolved from D to 2.6D with increasing flow conditions. In this domain, the Darcy-Weisbach resistance coefficient f can be approximated using a constant for a given slope. The results using this new flow resistance equation proved to be more accurate than those using equations obtained from simple fittings of logarithmic laws to mean values. The data set indicates that distinguishing domains 2 and 3 is still relevant for bed load. In particular, the data indicate a slope dependence in domain 2 but not in domain 3. A bed load model, based on the tractive force concept, is proposed. Finally, flow resistance and bed load equations were used together to calculate both shear stress and bed load from the flow discharge, the slope, and the grain diameter for each run of the data set. Efficiency tests indicate that new equations (implicitly taking a feedback mechanism into account) can reduce the error by a factor of 2 when compared to other equations currently in use, showing that feedback between flow resistance and bed load can improve field bed load modeling.