# Influence of Right Triangular Prism Rough Beds on Hydraulic Jumps
## 3 Results and Analysis
### Effects of Right Triangular Prism Rough Beds on Sequent Depth Ratio  
The conjugate depth \(y_{2}\) is the basis for defining the stilling basin side wall's height to help reduce the downstream flow influence on the riverbank. Through Table 7, the values of the conjugate depth ratio \(y_{2}/y_{1}\) were calculated for all numerical cases. Based on Equation (2) and the multiple linear regression method (MLR), this study found a relationship between the sequent depth ratios, the shape factors of the right triangular prism rough beds, and inflow conditions (see Equation (12)).  
\[\frac{y_{2}}{y_{1}}=1.13 Fr_{1}-0.2\frac{t}{y_{1}}-0.162\frac{L_{s}}{y_{1}}-0.19 4,R^{2}=0.987 \tag{12}\]  
Equation (12), which obtained a very high correlation coefficient of 0.987, indicated that the \(y_{2}/y_{1}\) predicted results agreed with the \(y_{2}/y_{1}\) value from numerical simulation results (error \(<5\%\)), as shown in Figure 8a. From Equation (12), the \(y_{2}/y_{1}\) values were strongly dependent on \(Fr_{1}\) (see Figure 8b), which is understood when supercritical flow changes to subcritical flow through the hydraulic jump phenomenon. On the contrary, the two parameters (\(t/y_{1}\) and \(L_{s}/y_{1}\)) decrease with the sequent depth ratio, as shown in Figure 9. According to MLR analysis results, the upstream and downstream slopes of triangular roughness (\(m_{1,2}\)) have a minimal effect on the increase in sequent depth ratio. Thus, it is absent in Equation (12). This could be explained by the variation in the flow field in the hydraulic jump zone and the small eddy structures that appear within the rough bed range (analyzed in detail in Section 3.7).  
Figure 10a plots this study's depth ratios (\(y_{2}/y_{1}\)) and earlier studies versus \(Fr_{1}\). This figure shows that the depth ratio \(y_{2}/y_{1}\) for the rough beds is smaller than the smooth bed (classical jump). Therefore, the effect of the right triangular prism rough beds reduces in comparison with the conjugate depth. The results of this study also indicate a good agreement with those obtained by previous studies [1,2,5,7,8,13].  
This research uses the \(D=(y_{2^{*}}-y_{2})/y_{2^{*}}\) parameter introduced by [PERSON] and [PERSON] [1] to define the reduction in the conjugate depth between rough and smooth beds. The variation in the \(D\) parameter against \(Fr_{1}\) is shown in Figure 10b. The \(D\) parameter has an average value of 0.217, indicating that hydraulic jumps on the rough beds provide a conjugate depth after the jump that is approximately 21.7 % less than the smooth bottom.  
On the other hand, Figure 10b also shows that the M_III and VI models effectively reduce the conjugate depth better than the M_I and II models. The maximum D value is 0.287 for the M_III model with \(Fr_{1}=5.0\), and the minimum is 0.171 for the M_I model with \(Fr_{1}=5.65\). The findings of this study are in good agreement with those of [PERSON] and [PERSON] [1], [PERSON] [7], and [PERSON] and [PERSON] [8]. They are about 0.25, 0.2, and 0.2, respectively.  
Figure 7: Normalized water surface profiles for the hydraulic jump region on rough bed models.