Flow Resistance and Hydraulic Geometry in Gravel-And Boulder-Bed Rivers

Abstract

The frictional resistance of river beds affects how water discharge is partitioned between depth and velocity, which is important in many aspects of hydrology, geomorphology, and aquatic ecology. Many of the most widely-used resistance equations predict reach-average velocity from relative submergence (RS), the ratio of mean flow depth to a bed roughness height such as the 84th percentile of the bed grain-size distribution (D₈₄). Nondimensional hydraulic geometry (HG) is an alternative approach that directly partitions unit discharge into depth and velocity. We show that any RS equation has an implicit or explicit HG equivalent, and the other way round. Analysis of a large set of flow measurements in gravel- and boulder-bed channels confirms previous findings that HG equations using D₈₄ outperform mathematically equivalent RS equations in predicting velocity. This paradox is explained by mathematical analysis and numerical experiments, both of which show that HG equations are less sensitive to the inevitable measurement uncertainty in the variables required for a prediction and the observed velocity used for testing. We also propose a new, simple and effective HG equation using D₈₄ to predict depth and velocity from unit discharge. It is derived in the same way as the now widely-used variable-power equation equation (Ferguson, 2007, https://doi.org/10.1029/2006wr005422) and for deep flows it reduces to an inverted Manning-type equation. It should be possible to use HG equations for flow resistance in sand-bed and bedrock rivers, but this may require new definitions of roughness height.