# Influence of Right Triangular Prism Rough Beds on Hydraulic Jumps
## 3 Results and Analysis
### Evaluate Variation in the Bed Shear Stress for Rough Beds  
Increased bed shear stress is one of the primary reasons for the reduction in conjugate depth and the length of the hydraulic jump over the rough beds. The shear force coefficient \(\varepsilon_{f}\)[1] mentioned in this section was determined as follows:  
\[\varepsilon_{f}=\frac{P_{2}-P_{1}+M_{2}-M_{1}}{\gamma y_{1}^{2}/2} \tag{16}\]  
where \(P_{1}\), \(P_{2}\), \(M_{1}\), and \(M_{2}\) are the hydrostatic pressure force and momentum flux per unit width at the sections before and after the jump, respectively [22].  
Based on the numerical model results data, as shown in Table 7, this study calculated the \(\varepsilon_{f}\) values by Equation (16). We used the MLR method to find a relationship between the \(\varepsilon_{f}\) parameter, geometry characteristics of rough beds, and inflow conditions, which is written below.  
\[\varepsilon_{f}=9.208 Fr_{1}+2.549\frac{t}{y_{1}}+2.06\frac{L_{s}}{y_{1}}-36.335,R^{2}=0.978 \tag{17}\]  
The \(\varepsilon_{f}\) predicted values calculated by Equation (17) are compared with the \(\varepsilon_{f}\) numerical simulation result values, as shown in Figure 18a. This figure shows that the \(\varepsilon_{f}\) values calculated by Equation (17) entirely agree with the numerical result values with an error of \(<\)10%. Equation (17) demonstrates that the \(\varepsilon_{f}\) values are strongly dependent on \(Fr_{1}\) parameters. On the contrary, the two parameters (\(t/y_{1}\), \(L_{s}/y_{1}\)) have a negligible effect on increasing the bed shear stress, as shown in Figure 19.  
In Figure 20a, the shear force coefficient \(\varepsilon_{f}\) was plotted versus inlet Froude numbers \(Fr_{1}\) for other works. This figure shows good agreement between this study's and previous researchers' results. On the other hand, Figure 20b shows that the number of \(\varepsilon_{f}\) hydraulic jumps over the right triangular prism rough beds is almost ten times that of the bed shear stress in jumping over the smooth bed, and it can be written by the equation below:  
\[S\varepsilon_{f}=101.18 Fr_{1}{}^{-1.25},R^{2}=0.844 \tag{18}\]  
Figure 17: Variation in the \(\eta\) energy loss parameters versus \(Fr_{1}\) for all rough bed models.  
Figure 19: The \(\varepsilon_{f}\) values versus \(t/y_{1}\) and \(Ls/y_{1}\) for all rough bed models.  
Figure 20: (**a**) Variation in the \(\varepsilon_{f}\) values with \(Fr_{1}\) for other works, (**b**) Variation in the surplus of shear force factor \(Se_{f}\) with \(Fr_{1}\)[1; 2; 4; 5; 8].  
Figure 18: (**a**) Predicted versus numerical results of \(\varepsilon_{f}\), (**b**) \(\varepsilon_{f}\) parameter against \(Fr_{1}\).  
The values of the shear force coefficient surplus parameter \(Se_{f}=(\varepsilon_{f}-\varepsilon_{f^{*}})/\varepsilon_{f^{*}}\) versus the inlet Froude numbers are shown in Figure 20b. This figure shows that the maximum value of \(Se_{f}\) is equal to 17.3 with \(Fr_{1}=5.0\) for M_III model, and the minimum value of \(Se_{f}\) is equal to 5.5 with \(Fr_{1}=9.4\) for M_I model. This research indicates that the value of \(\varepsilon_{f}\) is more significant when the models are placed with a discontinuous roughness element on the channel bed compared to when they are arranged with a continuous roughness element. Additionally, the shear force coefficient of the rough bed models analyzed in this research is about ten times greater than that of the smooth bed models under the same test conditions. (\(\varepsilon_{f^{*}}\) is the shear force coefficient of a classical jump).