## 2 Experiments and Analysis
### Experimental Results and Range Analysis
#### 2.2.2 Effect of Test Factors on Erosion Depth  
Table 3 shows the range analysis of the erosion depth of the submerged water jet. Range analysis can determine the influence order of each factor on test results, and obtain the possible optimal level of each factor to design a better test condition. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**Levels**} & \multicolumn{4}{c}{**Factors**} \\ \cline{2-5}  & \(\boldsymbol{p}\) **(MPa)** & \(\boldsymbol{\alpha}\) (\({}^{\circ}\)) & \(\boldsymbol{t}\) **(s)** & \(\boldsymbol{s}\) **(mm)** \\ \hline
1 & 2 & 30 & 20 & 10 \\
2 & 4 & 45 & 40 & 15 \\
3 & 6 & 60 & 60 & \\
4 & 8 & 90 & 80 & \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Factors and levels of orthogonal experimental design. Figure 4: Erosion test parameters. (**a**) Erosion depth \(H\) (sectional view), (**b**) jet pressure \(p\), jet angle \(\alpha\), standoff distance \(s\). \begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Test No.**} & \multicolumn{4}{c}{**Factors**} & \multicolumn{2}{c}{**Evaluation Indexes**} \\ \cline{2-5}  & \(\boldsymbol{p}\) **(MPa)** & \(\boldsymbol{\alpha}\) (\({}^{\circ}\)) & \(\boldsymbol{t}\) **(s)** & \(\boldsymbol{s}\) **(mm)** & \(\boldsymbol{H}\) **(cm)** \\ \hline
1 & 2 & 30 & 20 & 10 & 0.087 \\
2 & 2 & 45 & 40 & 10 & 0.194 \\
3 & 2 & 60 & 60 & 15 & 0.336 \\
4 & 2 & 90 & 80 & 15 & 0.306 \\
5 & 4 & 30 & 40 & 15 & 0.412 \\
6 & 4 & 45 & 20 & 15 & 0.468 \\
7 & 4 & 60 & 80 & 10 & 0.906 \\
8 & 4 & 90 & 60 & 10 & 0.582 \\
9 & 6 & 30 & 60 & 10 & 1.112 \\
10 & 6 & 45 & 80 & 10 & 1.556 \\
11 & 6 & 60 & 20 & 15 & 1.104 \\
12 & 6 & 90 & 40 & 15 & 1.088 \\
13 & 8 & 30 & 80 & 15 & 1.906 \\
14 & 8 & 45 & 60 & 15 & 2.108 \\
15 & 8 & 60 & 40 & 10 & 2.068 \\
16 & 8 & 90 & 20 & 10 & 1.702 \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Results of the hybrid orthogonal test L\({}_{16}\) (4\({}^{3}\times\)2\({}^{1}\)). \(R\) is the range, calculated by Equation (1), which reflects the fluctuation of test results and the significance of the corresponding factor's effect on the evaluation index. A larger \(R\) indicates that a change in this factor will cause a larger change in test results. This means that the influence of this factor on the evaluation index is greater. The actual definitions are:  
\[\left\{ \begin{array}{l}R=k_{\text{max}}-k_{\text{min}}\\ k_{i}=\frac{K_{i}}{r}\qquad\quad i=1,2\ldots q\\ K_{i}=\sum\limits_{j=1}^{r}y_{ij}\qquad i=1,2\ldots q\\ r=\frac{1}{q}\end{array} \right. \tag{1}\]  
where \(k_{i}\) is the average value of the indexes, which represents the average value of test results of the \(i\) level of the factor, and reflects the comprehensive effect of the \(i\) level on the evaluation index. \(k_{\text{max}}\) and \(k_{\text{min}}\) correspond to the maximum and minimum values of \(k_{i}\), respectively. \(K_{i}\) is the sum of the indexes, which represents the sum of the results of the \(i\) level of the factor. \(y_{ij}\) corresponds to the index value \(H\) of the \(j\) test in the \(i\) level. \(i\) represents the number of the level. \(r\) is the number of repeated tests at the \(i\) level of the factor. \(j\) is the number of the test from 1 to \(r\). \(n\) is the total number of tests. \(q\) is the total number of levels of the factor. In the hybrid orthogonal table, the different number of levels of each factor will lead to a greater \(R\) for the factor with more levels, although this factor may have the same impact as other factors. Therefore, \(R^{\prime}\) is used to correct the range \(R\) for the factors with different levels [40], as shown in Equation (2), and the correction coefficients \(d\) are shown in Table 4.  
\[R^{\prime}=d\times\sqrt{r}\times R. \tag{2}\]  
As shown in Table 3, \(R^{\prime}{}_{p}>R^{\prime}{}_{t}>R^{\prime}{}_{a}>R^{\prime}{}_{s}\). Therefore, jet pressure has the most significant effect on erosion depth. Impact time and jet angle follow closely, and their correction ranges are relatively