Flow resistance in gravel-bed rivers: Progress in research

Abstract

Resistance to flow determines several important hydraulic parameters in streams and rivers and must be properly represented in models for estimating water discharge and sediment transport. This paper reviews our understanding of how flow resistance is generated in open channel flows and evaluates the different approaches used to model the flow resistance that originates at the bed in coarse-grained alluvial rivers. The Manning equation is routinely used as a flow predictor in hydraulic models of open channel flow. However, defining the energy gradient and roughness coefficient can be problematic and as a result, estimates of velocity and discharge are subject to considerably uncertainty and error. Attempts to develop more physically based models of flow resistance exploit well-established principles of engineering fluid mechanics. Forces responsible for generating resistance at the channel bed are of two kinds—shearing forces and pressure forces. The former are generated by transfers of fluid momentum and give rise to skin friction. The latter are generated by pressure gradients around roughness elements and give rise to form drag. The relative importance of skin friction and form drag varies with the relative submergence—the ratio of the flow depth (Y) (or hydraulic radius (R)) to the size of the bed material (D). Skin friction dominates at high relative submergence (Y≫D) and models for such conditions are based on boundary layer theory and the results of its application to engineering studies of pipe flow. Models assume a logarithmic velocity profile through the depth and have the form 1/√ff∝log10 (R/ks) where ff is the Darcy–Weisbach roughness coefficient and ks is a bed roughness parameter scaled on bed material size (ks≈3.5D84 where D84 is the surface size for which 84% is finer than). Such models can be approximated by a power equation of the form 1/√ff∝(R/D)b with an exponent of 1/6. The assumption of a logarithmic velocity profile is invalid for low relative submergence conditions (Y≈D). For these flow conditions, flow resistance models are based on either hydraulic geometry or roughness layer theory. The equations that result from the two approaches have been shown to be equivalent and, moreover, to be approximated by 1/√ff∝(R/D). Since this, in turn, is equivalent to a power law with an exponent of 1, a variable power relation which is asymptotic to the logarithmic flow law and roughness layer laws at high and low relative submergence respectively provides a single equation that can be used in both deep and shallow flows as it explicitly accounts for the changing sources of flow resistance as relative submergence changes. A similar approach using dimensionless hydraulic geometry yields an equation with different parameter sets for deep and shallow flows. Assessments of the predictive performance of a variety of flow resistance equations suggest that the dimensionless hydraulic geometry equation for low-high relative submergence performs best, closely followed by the logarithmic flow law and the variable power flow law. Predictive errors, however, can be significant and further research is needed to better understand the physics of flow over rough boundaries and to incorporate this understanding in improved models of flow resistance.