Experimental Study of the Effect of Bed Forms on Darcy-Weisbach Friction Coefficient in Straight Open Channels

Introduction: Among the most important parameters in hydraulic engineering is the flow resistance coefficient, e.g. Darcy-Wiesbach, in alluvial rivers; in which is divided into two categories: grain resistance and form resistance. The grain resistance is a function of the bed sediment size; whilst the form resistance is a function of the bed form geometry. Ripple and dune are among the most common forms formed in alluvial rivers that are triangular in shape (Shafai Bajestan, 2008; Julien, 2010). The effect of bed form on flow resistance have been studied by few researchers such as: Talebbeydokhti et al. (2006), Omid et al. (2010), Nasiri Dehsorkhi et al. (2011), Chegini and Pender (2012), Kabiri et al. (2014), and Kwoll et al. (2016). However, the effect of roughened bed form, bed form covered with artificial roughness of different sizes, on the Darcy-Wiesbach friction coefficient
has received little attraction. Therefore, it is the main goal of the present study to experimentally investigate this effect. This different flow rates and different bed slopes. In this study, sediments with sizes of 0.51 and 2.18 mm were used to rough the surface of the bed forms.

Methodology: The experiments were performed in a sloping straight flume (manufactured by Armfield, UK). The length and width of the flume were 12 and 0.3 m, respectively (Figure 1). In this study, flow rates of 10, 15, 20, 25, and 30 l/s and bed slopes of 0, 0.0001, 0.0005, 0.001, and 0.0015 were examined.
The present study experiments were divided into two categories: bed without form and bed with form. Each form was made by P.V.C sheet in a triangular shape. The bed form length and height were equal to 20 and 4 cm, respectively, and the angles of its upstream and downstream to the horizon were selected as 16.4 and 32 degrees, respectively. After making each form, the desired sediments were glued on their surface. In this study, two types of uniform granulation with average sizes (d50) of 0.51 and 2.18 mm were used. The total number of experiments in the present study was 100.

Results and discussion: Ripple and dune form are usually being developed in lower flow regime, in which the Froude number is less than 1 (Shafai Bajestan, 2008; Julien, 2010). In this study, the Froude number values in all tests with bed form were from 0.435 to 0.6, indicating a lower flow regime.
Figure 5 shows the changes in the total Darcy-Weisbach's coefficient (f_b) against Froude number for sediment-covered with sand sizes of 0.51 and 2.18 mm. It can be seen that as the Froude number increased, the total Darcy-Wiesbach's coefficient (f_b) decreased. In addition, increasing the longitudinal slope of the bed, causes the f_b to increase. In addition, increasing the longitudinal slope of the bed, causes the f_b to increase.
Figure 6 shows the trend of changes in the total Darcy-Weisbach's coefficient (f_b) versus relative submergence (y/∆) for the slope of 0.0001. This figure shows that with increasing relative submergence rate, the total Darcy-Wiesbach's coefficient decreased due to the relative roughness reduction. In addition, Figure 6 shows that f_b increased with increasing particle size. Calculations showed that the value of f_b in beds with a sediment size of 2.18 mm for slopes of 0, 0.0001, 0.0005, 0.001, and 0.0015 on average 32.8, 28.8, 28.46, 33.8, and 35.9% are more than the bed covered with 0.5 mm sediment size, respectively.
The results of Table 2 shows that the grain Darcy-Weisbach's coefficient ((f_b ) ́) for particles with sizes of 0.51 and 2.18 mm are on average 25.45 and 26.8% of the total friction coefficient (f_b), respectively. In addition, from Darcy-Weisbach's coefficient (f_b^'') for particles with sizes of 0.51 and 2.18 mm is on average 74.55 and 73.2% of the total friction coefficient (f_b), respectively. According to the results, it can be seen that the value of f_b^'' for particles with a size of 0.51 and 2.18 mm on average is 193.6 and 173.4% more than (f_b ) ́, respectively.

Conclusion: The results showed that with increasing the particle size of the bed, the total Darcy-Wiesbach's coefficient (f_b) and the grain Darcy-Wiesbach's coefficient ((f_b ) ́) increased. The value of f_b in sedimentary beds with a size of 2.18 mm is on average 32% higher than sedimentary beds with a size of 0.5 mm. Meanwhile, the value of the form Darcy-Wiesbach's coefficient (f_b^'') for particles with a size of 0.51 and 2.18 mm on average is 193.6 and 173.4% more than (f_b ) ́, respectively.

Keywords: Bed form, Ripple, Dune, Darcy-Weisbach friction coefficient.