Mixed scaling for open-channel flow over gravel and cobbles

Mixed scaling that includes inner and outer variables is performed on open-channel flow over gravel and cobbles. A similarity analysis is performed following classification of the flow as an equilibrium turbulent boundary layer using the asymptotic invariance principle. A mixed scale that accounts for the variation of turbulent length in the vertical direction is used to parameterize the external boundary impact on the flow for the logarithmic and outer layers. The resultant scale uses free-stream velocity, hydraulic radius, and bed particle size in which 84% of the sediment is finer. The results are consistent with empirical gravel-bed river research and applications, and nicely include the variation of the turbulent flow structure into the scaling. The results are verified using data from gravel- and cobble-bed flumes and rivers and by comparison with earlier methods. A single coefficient, semitheoretical model collapses laboratory and field data well. An integral approach is applied to allow the asymptotic invariance principle equilibrium to be more applicable in future practice.