Relatively Rough Flow Resistance Equations

Abstract

The definitions of depth and hydraulic radius become ambiguous when bed roughness is large relative to flow depth. Various statistics are currently used to describe bed roughness and many different flow resistance formulas have been developed. The volumetric hydraulic radius $R_v$ and the standard deviation of bed surface elevations $d_z$ are rational and unambiguous measures suitable for large relative roughness conditions. Their influence on flow resistance is investigated using conceptual models and digital elevation models of natural alluvial beds. The results show that head-losses for large-scale relative roughness beds can be related to $(R_v/d_z);$ the $(R_v/d_z)$ exponent of power-law flow resistance equations increases from 1/6 to more than 1/2 as relative roughness increases, and flow velocity can be determined from boundary topography measures, water level and slope, without any calibrated coefficients. An overlooked form of the log law, using standard deviation $d_z\text{,}$ performs as well as power laws for predicting flow resistance with high relative roughness and it reverts to the conventional log law when relative roughness is low. A field technique for determining $R_v$ and $d_z$ is described.