## 4 Discussion  
Cluster analysis is a mathematical statistical method that categorizes research objects based on the information provided [38]. Its essence lies in classifying data by proximity,  
Figure 5: (**a**) Gas holdup distribution under different injection pressure conditions; (**b**) the distribution of gas holdup at the central axis of the jet; (**c**) the horizontal distribution of the gas holdup at the outlet of the jet field. minimizing intra-category differences and maximizing inter-category differences. In multivariate statistical analysis, clustering can achieve dimensional reduction. The fundamental concept of hierarchical cluster analysis is to group variables based on proximity, starting with those closest in distance and progressively clustering more distant variables until each variable is appropriately assigned to a class. The cluster analysis is performed on 16 groups of sample data in this study. The basis for clustering is measured in terms of the distance between the individual samples. Each group of data can be regarded as a point in Rp (P-dimensional space), and n groups of sample data are n points in \(R^{p}\). Euclidean distance should be used to measure the distance between these n points, that is, the degree of proximity. Suppose \(X=(X_{1},\,X_{2},\,\ldots,\,X_{p})^{T}\) and \(Y=(Y_{1},\,Y_{2},\,\ldots,\,Y_{p})^{T}\) is a \(p\)-dimensional vector, which can be regarded as two points in the p-dimensional space. The Euclidean distance between the measurement points \(X\) and \(Y\) can be expressed as follows:  
\[d(x,y)=\left[\sum_{i=1}^{p}\left(X_{i}-Y_{i}\right)^{2}\right]^{0.5} \tag{9}\]  
In practical applications, the jet will form a certain coverage area upon reaching the water surface when the nozzle height above the water surface is 9 m (Figure 6). To optimize the nozzle parameters, the variation pattern of the cover width is studied based on cluster analysis using the operating conditions listed in Table 1. Due to duplicates in operating conditions 3, 6, and 11 in Table 1, these three conditions were merged to obtain a reorganized table of operating conditions, and the corresponding data results are shown in Table 3.  
Figure 7a illustrates the distances between diameter values calculated under different operating conditions. Figure 7b displays the categorization results based on these distances. From the analysis of individual operating conditions, it can be observed that they can be classified separately into three categories when the distance between operating conditions  
Figure 6: High-pressure jet schematic for red tide control. \begin{table}
\begin{tabular}{c c c c c} \hline \hline \multirow{2}{*}{**No.**} & **Initial Jet Flow** & \multirow{2}{*}{**Cone Angle**} & **Nozzle** & **Equivalent** \\  & **Pressure** & & **(0)** & **Diameter** & **Diameters** \\  & **(\(p\)/MPa)** & & **(\(d\)/mm)** & **(\(d\)/mm)** \\ \hline
1 & 0.36 & 105\({}^{\circ}\) & 2 & 39.3 \\
2 & 0.36 & 110\({}^{\circ}\) & 2 & 45.1 \\
3 & 0.36 & 115\({}^{\circ}\) & 2 & 54.8 \\
4 & 0.36 & 120\({}^{\circ}\) & 2 & 46.9 \\
5 & 0.36 & 115\({}^{\circ}\) & 1 & 41.2 \\
6 & 0.36 & 115\({}^{\circ}\) & 4 & 42.8 \\
7 & 0.36 & 115\({}^{\circ}\) & 6 & 40.6 \\
8 & 0.18 & 115\({}^{\circ}\) & 2 & 46.7 \\
9 & 0.27 & 115\({}^{\circ}\) & 2 & 49.1 \\
10 & 0.48 & 115\({}^{\circ}\) & 2 & 47.1 \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Numerical simulation results of equivalent diameters under different jet parameters. is less than 3. The first category is the working conditions 5, 6 and 7. The second category is working conditions 2 and 4; the third category consists of conditions 8, 9, and 10. The conclusion, derived from the clustering results in Figure 7b, indicates that the jet impact ranges obtained under these three categories exhibit good internal similarity. Working condition 1, with distance values ranging between 4.43 and 15.63 from the other conditions, is separated into a distinct category.