Flow resistance law under equilibrium bed-load transport conditions

The uniform flow resistance equation, in the form due to Manning or Darcy-Weisbach, is often applied to determine the stage-discharge relationship of a river cross-section. The application of this equation, namely the slope-area method, allows to indirectly measure by water level readings the corresponding river discharge. In this paper, a recently deduced flow resistance equation for open channel flow was tested during conditions of equilibrium bed-load transport. First the flow resistance equation was determined by dimensional analysis and applying the condition of incomplete self-similarity for the flow velocity profile. Then the analysis was developed by the following steps: (i) the flow resistance equation was calibrated by available laboratory measurements of flow velocity, water depth and bed slope carried out in 143 flume experimental runs during conditions of equilibrium bed-load transport; (ii) a relationship (Eq. (33)) between the Γ function of the velocity profile, the channel slope, the Shields number and the Froude number was established by the available measurements; (iii) the relationship for estimating the Γ function (Eq. (33)) and the theoretical flow resistance equation were tested by 127 independent flume measurements carried out for flows with bedload transport (iv) Eq. (33) and the flow resistance law were tested by field measurements carried out in 104 reaches of some Calabrian gravel bed rivers; and (v), finally, the relationship for estimating the Γ function was recalibrated (Eq. (34)) using all laboratory measurements (270 experimental runs). The proposed theoretical resistance law, calibrated by the measurements of Recking et al. (2008), allowed an estimate of the Darcy-Weisbach friction factor which is more accurate than that obtained by the original approach of Recking et al. The theoretical flow resistance law (Eq. (28)) coupled with the relationship for estimating the Γ function (Eq. (34)), which is characterized by the applicability of a wide range of flow conditions, allowed to estimate the Darcy-Weisbach friction factor for flows with and without bed-load.