Friction factor of river channel flows in rough steep slope mountaineous areas using energy slope division

Abstract: The corridor restoration of mountain river channels has been a considerable object over the past four decades. Among them, the friction factor of river channels can be one of the most important variables in the theory of stream corridor restoration. The Daycy friction factor is a function of the depth-to-sediment diameter (h/d) ratio for the traditional open channel flows. However, the friction factor formula is not applicable for the open channel flows on the rough steep slope mountain. The friction factor is dependent on both the h/d and energy slope S. In this study, an impervious flume with the adjustable slope was designed to investigate the river resistance of the steep slope mountain in the laboratory. The flume bottom was covered with uniform sediment in the diameter of 0.5, 1, and 1.85 mm. The flow discharge varied from 0.5 to 2.5 L/s. The h/d (divergence) was within the range of 0.84-7.27. Three slopes with the degree of 10°, 25°, and 35° were used to test the effect of the slope on the resistance factor. Experimental results show that the Darcy friction factor was larger than that for the traditional open channels, due to the roll waves on the water surface of the mountain rivers channels. The variable roll waves modified the flow resistance, leading to the stage-discharge relationship of the channel conveyance. Assume that the energy slope in a steep slope mountain channel was divided into two major components, i.e., the energy slope S1 without the roll waves, and the other S2 related to the roll waves. The energy slope S2 was caused by the roll waves that were created on the water's surface. The rolling wave occurred on the much larger slope of open channel. The Colebrook-White formula was used to calculated the energy slope without the roll waves S1 or related friction factor f1. An indirect empirical treatment was carried out to measure the roll wave with the energy slope S2. The formula of S2 and the related friction factor f2 were derived from the energy slope S2 using present experimental data. Among them, the energy slope S2 increased with the total slope S. Finally, a semi-analytical model was developed to compute the Darcy friction factor for the steep slope open channels. The total Darcy friction factor was set as the sum of two friction components, corresponding to the traditional open channel and roll wave resistance: f=f1+f2. A total of 48 datasets of measured flume data were selected to test the validity of the present energy division formula. Each data set included the complete records of flow discharge, channel width, water depth, energy slope, median sediment size, and specific gravity of sediment. The comprehensive database covered a wide range of slopes in the mountain river channels. A comparison was made between the present formula with the measured data. Nearly all the data lay within the ±20% error band, indicating excellent consistency. Furthermore, the present formula was also developed on the basis of the mechanism of steep slope mountain river energy dissipation. The finding can provide a fundamental theory for the mountain river.