At-a-site variation and minimum flow resistance for mountain rivers

Abstract

Existing flow resistance relationships for mountain rivers may be in error by typically 30%. In part this is because flow resistance is a complex phenomenon, not easy to quantify with the available data. However, a further reason is the reliance of past analyses on fitting relationships to data from multiple sites, hence emphasizing an averaging between-site variation. A more scientific approach is to derive separate at-a-site relationships and investigate how they can be collapsed into a single formula. Twenty-seven carefully selected field datasets from seven literature sources are therefore analysed empirically to show at-a-site variations in resistance, with the aim of improving the accuracy and applicability of mountain river flow resistance relationships. Considering only the bed grain roughness, the principal factors likely to determine flow resistance are identified as relative submergence d/ D 84 (where d is depth and D 84 is the bed material 84-percentile size), bed material size distribution and channel slope. The dependency of the resistance function (8/ f) 1/2 (where f is the Darcy-Weisbach resistance coefficient) on d/ D 84 is more accurately described by a power law than by a semi-logarithmic law, especially at high flows. From observations of coincident at-a-site datasets, two equations are derived, distinguished largely according to channel slope S. For S<0.8%, (8/ f) 1/2=3.84( d/ D 84) 0.547 ( r 2=0.986). For S>0.8% (8/ f) 1/2=3.10( d/ D 84) 0.93 ( r 2=0.959). These equations appear to define minimum values for flow resistance, probably for ideal conditions of bed grain roughness and uniform flow. Deviations from this ideal (because of form roughness, nonuniform flow or bank effects) will increase resistance in a manner which is not yet quantified. The equations therefore have a very specific applicability and are not generally applicable to all flows. The cause of the slope dependency remains unclear, nor is any dependency on bed material size distribution identified. Nevertheless the high correlation coefficients suggest an improved accuracy compared with other field-based relationships. The equations are derived for in-bank flows with d/ D 84<11 and slopes in the range 0.2–4%.