# Influence of Right Triangular Prism Rough Beds on Hydraulic Jumps
## 2 Mathematical Method
### Define Computational Grid and Boundary Conditions for Numerical Simulations  
In this research, the calculated domain of the numerical models is made up of two mesh blocks (see Figure 2). The contained mesh block was set up for the whole numerical model space; meanwhile, the rough bed region is set by the nested mesh block because these regions need a finer mesh to capture the flow behavior better. A structured grid with the same three-dimensional grid size was used in these numerical models. This study also considers three computational grids (see Table 2) to evaluate the calculated grid independence for the numerical result, determine the appropriate grid size for the best steady numerical results, and apply this grid to the subsequent simulations. \begin{table}
\begin{tabular}{l c c c} \hline \hline \multirow{2}{*}{**Mesh Name**} & \multicolumn{2}{c}{**The Cell Size of**} & \multicolumn{2}{c}{**The Cell Size of**} & \multirow{2}{*}{**Node**} \\  & **Nested Mesh Block** & **Contained Mesh Block** & & \\  & \(\Delta\mathbf{x}=\Delta\mathbf{y}=\Delta\mathbf{z}\) (mm)** & \(\Delta\mathbf{x}=\Delta\mathbf{y}=\Delta\mathbf{z}\) (mm)** & & \\ \hline Grid 1 & 2.0 & 4.0 & 4.4 \(\times\) 10\({}^{6}\) (fine) \\ Grid 2 & 2.5 & 5.0 & 2.6 \(\times\) 10\({}^{6}\) (medium) \\ Grid 3 & 3.5 & 7.0 & 1.2 \(\times\) 10\({}^{6}\) (coarse) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Characteristics of the computational grids. Based on the experimental data reported by [PERSON], [PERSON] et al. [4], the boundary condition of the numerical models was set up as shown in Figure 2: The inlet boundary condition (upstream channel) simultaneously assigns velocity \(V_{1}\) and supercritical depth \(y_{1}\). The outlet boundary is set at the water level with a subcritical depth of \(y_{2}\). The free surface boundary condition provides the atmospheric pressure. The channel bottom is a wall boundary, and symmetry boundary conditions provide the inner boundary conditions. This paper uses the grid convergence index (_GCI_) method, which is introduced and recommended by [PERSON] and [PERSON] [21], to evaluate calculated grid independence, as described in Equation (11). \[GCI_{i+1;i}=\frac{1.25\left|\frac{\phi_{i}-\phi_{i+1}}{\phi_{i}}\right|}{\left( \frac{h_{i+1}}{h_{i}}\right)^{p}-1} \tag{11}\]  
where the value of \(p\) is equal to 10.76 and is called the accuracy local order; \(\phi_{i},h_{i}\) are the calculation result and grid size of the \(i\)-th grid, respectively, and satisfy the relationship of \(h_{1}<h_{2}<h_{3}\); and \(i\) was taken by 1 and 2 only. The \(\phi_{i}\) variable is used to judge mesh convergence. The \(\phi_{i}\) variable value is assigned to \(y_{2}/y_{1}\) (sequent depth ratio), which changes with different calculated meshes. In this research, we use the numerical result of the rough bed M_II model for the No.82 test (with initial conditions such as \(y_{1}=0.02\) m and \(Fr_{1}=5.65\)). The results of the mesh convergence analysis are shown in Table 3. In this table, the \(GCI_{21}\) value equals 0.031 percent, which is considered very small. On the other hand, to verify that the numerical results are unique and accurate, a suitable mesh size needs to be chosen. This article compares the \(L_{j}/y_{1}\) jump length ratio values obtained from two types of grids, Grid 1 and Grid 2. Table 4 shows the results of the jump length ratio (\(L_{j}/y_{1}\)) between Grid 1 and Grid 2, with errors less than 5%. Therefore, Grid 1 and Grid 2 almost satisfy the grid's independence. Based on the efficiency, accuracy, and computation time of numerical simulation results, Grid 2 is used for all subsequent simulations. \begin{table}
\begin{tabular}{l c c} \hline \hline
**Parameters** & **Values** & **Note** \\ \hline \(\phi_{1}\) (-) & 6.110 & Grid 1 \\ \(\phi_{2}\) (-) & 6.125 & Grid 2 \\ \(\phi_{3}\) (-) & 6.725 & Grid 3 \\ \(P\) (-) & 10.76 & \\ \(GCI_{21}\) (\%) & 0.031 & \\ \hline \hline \end{tabular}
\end{table} 
Table 3: The mesh convergence analysis for calculated grids.